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ELEMENTS 


op 

SURVEYIN 


AND 


LEVELING. 


By  CHARLES  DAVIES,  L L.  D. , 

AUTHOR  OP  A FULL  COURSE  OP  MATHEMATICS. 

REVISED  BY 

J.  HOWARD  VAN  AMRINGE,  A.M.,  Pii. 

PROFESSOR  OP  MATHEMATICS  IN  COLUMBIA  COLLEGE. 


NEW  YORK  CINCINNATI  CHICAGO 

AMERICAN  BOOK  CO  M P A N Y 


DAVIES’S  MATHEMATICAL  SERIES 


For  Elementary  Schools 

Davies’s  Primary  Arithmetic 
Davies’s  Intellectual  Arithmetic 
Davies’s  First  Book  in  Arithmetic 
Davies’s  Standard  Arithmetic 
Davies’s  Practical  Arithmetic 
Davies’s  Complete  Arithmetic 

For  Secondary  Schools 

Davies’s  University  Arithmetic 
Davies’s  New  Elementary  Algebra 
Davies’s  Bourdon’s  Algebra 
Davies’s  Elementary  Geometry  and 
Trigonometry 

Davies’s  Legendre’s  Geometry  and 
Trigonometry 

For  Colleges  and  Advanced  Students 
Davies’s  University  Algebra 
Davies’s  Analytical  Geometry 
Davies’s  Analytical  Geometry  and 
Calculus 

Davies’s  Descriptive  Geometry 
Davies’s  Elements  of  Surveying 


Copyright,  1870,  by  Charles  Davies 
Copyright,  1883,  by  A.  S.  Barnes  & Co. 
Copyright,  1898,  byj.  Mansfield  Davies 

davies’s  surveying 
w.  p.  5 


- 


PREFACE. 


AVIES’  Elements  of  Surveying,  first  published  in  1830, 


I J was  designed  as  a text-book  for  the  pupils  of  the  XL  S. 
Military  Academy,  and  in  its  preparation  little  regard  was  had  to 
the  supposed  wants  of  other  institutions. 

The  work  was,  however,  received  by  the  public  with  more 
favor  than  was  anticipated,  and  soon  became  a leading  text-book 
in  Colleges,  Academies,  and  the  higher  grade  of  Schools.  For 
the  purpose  of  adapting  it  more  fully  to  the  requirements  of 
these  institutions,  the  author  made  many  changes  in  successive 
editions,  and  gave  it  his  final  revision  in  1870. 

In  the  present  edition,  while  the  admirable  features  which 
have  hitherto  commended  the  work  so  highly  to  institutions  of 
learning  and  to  practical  surveyors  have  been  retained,  some  of  the 
topics  have  been  abridged  in  treatment  and  some  enlarged,  others 
have  been  added,  and  the  whole  has  been  arranged  in  the  order  of 
progressive  development. 

It  has  been  the  intention  to  begin  with  the  very  elements  of 
the  subject  and  to  combine  those  elements  in  the  simplest  man- 
ner, so  as  to  render  the  higher  branches  of  Plane'  Surveying  com- 
paratively easy.  The  necessary  principles  of  logarithms  and 
plane  trigonometry  are  given,  and  their  mode  of  application 
shown.  All  the  instruments  needed  for  plotting  have  been 
carefully  described ; and  the  uses  of  those  required  for  the 
measurement  of  angles  are  fully  explained. 

In  the  section  on  Magnetic  Declination  or  Variation  of  the 
Needle,  papers  of  the  U.  S.  Coast  and  Geodetic  Survey  have  been 
largely  used.  From  them  have  been  taken: — tables  of  annual 
changes  in  declination,  and  for  computing  the  declination  at  any 
epoch,  at  various  places  in  the  United  States,  which  will  be  found 


IV 


PREFACE. 


of  especial  value  in  re-running  lines  of  old  surveys;  also,  new 
tables  of  the  times  and  azimuths  of  Polaris  when  at  elongation, 
for  use  in  determining  the  true  meridian  with  compass  or  tran- 
sit, which  with  the  rules  given  for  interpolation  are  more  accurate 
than  any  similar  tables  previously  published. 

A full  account  is  given  of  the  system  adopted  in  the  survey  of 
the  public  lands;  and  although  the  method  is  simple,  it  has, 
nevertheless,  been  productive  of  great  results,  by  defining,  with 
mathematical  precision,  the  boundaries  of  lands  in  the  new  States, 
and  thus  settling  their  titles  on  an  indisputable  basis.  In  this 
connection  official  instructions  and  diagrams  issued  by  the  U.  S. 
General  Land  Office  have  been  used,  and,  as  the  principal  lines  of  a 
government  survey  must  be  run  with  reference  to  the  true  meridian, 
the  Solar  Compass  and  solar  attachment  to  transit  are  described. 

A change  made  in  the  present  edition,  which  must  prove  par- 
ticularly acceptable,  is  the  transformation  of  the  article  on  Mining 
Surveying  into  a complete  treatise,  in  which  the  location  of  claims 
on  the  surface,  the  latest  and  best  methods  of  underground  travers- 
ing, etc.,  the  calculation  of  ore-reserves,  and  all  that  pertains  to  the 
work  of  the  Mining  Surveyor,  are  fully  explained,  and  illustrated 
by  practical  examples.  This  improvement  is  due,  substantially,  to 
John  G.  Murphy,  Esq.,  E.M.,  at  one  time  Territorial  Geologist  of 
Wyoming  Territory,  an  expert  Mining  Engineer  of  large  and 
varied  practice. 

In  addition  to  acknowledgments  elsewhere  made,  the  under- 
signed is  indebted  to  Professor  Rodney  G.  Kimball,  of  the  Brook- 
lyn Polytechnic  Institute,  for  valuable  suggestions  and  labor,  and 
to  the  Messrs.  W.  and  L.  E.  Gurley,  of  Troy,  N.  Y.,  for  their 
courtesy  in  furnishing  him,  for  use,  advance  sheets  of  the  last 
edition  of  their  Instrument  Manual  and  many  cuts  of  their 
surveying  instruments. 

J.  H.  VAN  AMRINGE, 

Editor  of  Davies’  Course  of  Mathematics. 


Columbia  College,  N.  Y., 
t September , 1883. 


CONTENTS 


BOOK  I. 

INTRODUCTORY  PRINCIPLES  AND  DEFINITIONS. 


SECTION  PAGE 

I. — Logarithms 9-16 

II.— Plane  Trigonometry.... 17-32 

III.  — Instruments  for  Plotting.  33-39 

IV.  — General  Definitions 40-41 


BOOK  II. 

CHAIN  SURVEYING. 

42-58 
42 
44 
48 


58-70 


I. — Measurement  of  Distances  . . . 

Necessary  instruments 

To  measure  a horizontal  line. . 

Applications 

Standard  of  measure 

II. — Area  or  Contents  of  Ground. 


BOOK  III. 

COMPASS  SURVEYING. 


I. — Definitions 71-73 

II. — Surveyor’s  Compass 74-78 

III. — Work  on  the  Field 79-90 

Field  notes 79 

Necessary  measurements  on  the  field 80 

Errors  of  compass 80 

Measurements  by  off  set  courses  82 

General  example 84 

To  correct  local  attraction 89 


VI 


CONTENTS. 


SECTION  PAGE 

IV.  — Area  or  Contents  of  Ground 90-131 

Traverse  table  and  its  uses 90 

Balancing  the  work 94 

Double  meridian  distances 97 

Area 99 

Plotting 102 

Examples 104 

To  supply  omissions  \n  field  notes Ill 

Another  mode  of  finuing  areas 120 

V.  — Magnetic  Declination  or  Variation  of  the  Needle.  . .131-152 

Definitions 131 

Daily  variatior  (with  table) 132 

Secular  variation  (with  tables) 134 

Method  of  finding  declination  (with  tables) 140 

To  find  true  meridian  with  compass 145 

Vernier  compass  and  its  use 150 

Form  of  survey  bill 152 

BOOK  IV. 

TRANSIT  SURVEYING. 

I. — Surveyor’s  Transit 153-165 

II.— Measurement  of  Angles 166-183 

Horizontal  and  vertical  angles 166, 167 

Azimuths  and  bearings 168, 169 

To  find  true  meridian  with  transit 172 

Applications  to  heights  and  distances 174 

III.  — Ranging  out  Lines,  Etc 184-189 

To  range  out  a line 184 

Measurement  of  distances  by  transit 186 

To  survey  a road,  boundary  of  estate,  etc 189 

To  survey  streets  of  a town  or  city 189 

IV.  — Farm  Surveying  by  Transit 190-195 

BOOK  V. 

LAYING  OUT  AND  DIVIDING  LAND. 

I.— Of  Dividing  Land 196-203 

II. — Public  Lands  of  tiie  United  States . .203-211 


CONTENTS.  vii 

BOOK  VI. 

TRIGONOMETRICAL  SURVEYING. 

SECTION  PAGE 

I.— Making  the  Survey 212-232 

Definitions  and  general  remarks 212 

Base  line 215 

Signals 217 

Heliotrope 218 

Theodolite 219 

Measurement  of  angles,  and  notes 22 1 

Reduction  to  the  centre 226 

Survey  of  a harbor 228 

II. — Filling  up  the  Survey 233-240 

By  the  compass 233 

By  the  plane  table 234 

III. — Plotting  the  Triangulation 240-245 

BOOK  VII. 

LEVELING. 

I.— Definitions  and  Principles 246-248 

II. — Instruments 249-257 

Y Level 249 

Leveling  rods 255 

Tests  of  adjustment 256 

III.  — Leveling  in  the  Field 258-262 

IV. — Section  Leveling 263-276 

Definitions  and  principles 263 

Drawing  the  profile 270 

Establishment  of  the  grade 270 

Examples 276 

V.  — Cross-Section  Leveling 277-288 

Slopes 277 

Setting  slope  stakes 278 

VI.  — Computation  of  Earthwork 288-296 

BOOK  VIII. 

TOPOGRAPHICAL  SURVEYING. 

Definition  and  principles  297 

Examples  and  plotting 298 


CONTENTS. 


SECTION  pAGE 

Shading  and  delineation 310 

Topographical  signs,  U.  S.  Coast  Survey 313 

BOOK  IX. 

■RAILWAY  CURVES. 

Definitions  and  principles 317 

1 Location  of  curves 320 

Laying  off  ordinates  327 

Reversed  and  compound  curves 329 

BOOK  X. 

MINING  SURVEYING. 

I.— Definitions  and  General  Principles 338-340 

II. — Method  of  Locating  Claims 340-345 

III.  — Underground  Traversing,  Etc 346-361 

To  make  the  traverse 346 

To  reduce  the  traverse 350 

To  plot  traverse  on  the  surface 353 

To  plot  traverse  on  paper 360 

IV.  — Practical  Applications 361-374 

Problems 361 

Example  of  a developed  mine 366 

Calculation  of  ore-reserves 370 

Deposit  mine  with  mill  connections 372 

APPEND  ICES. 

A. — Solar  Compass  and  Solar  Attachment  to  Transit 1-15 

B. — Sextant 16-20 

C.  — Instructions  to  U.  S.  Mineral  Surveyors 21-29 

TABLES. 

Logarithms  of  Numbers 1-16 

Logarithmic  Sines  and  Tangents 17-62 

Natural  Sines 63-71 

Traverse  Table 72-161 


BOOK  I. 

INTRODUCTORY  PRINCIPLES  AND  DEFINITIONS. 


SECTION  I. 

LOGARITH  MS. 

1.  The  logarithm  of  a given  number  is  the  exponent  of  the 
power  to  which  it  is  necessary  to  raise  a fixed  number  to  produce 
the  given  number. 

Th q fixed  number  is  called  the  base  of  the  system.  In  the 
common  system,  to  which  alone  reference  is  made  in  this  section, 
the  base  is  10.  Every  number  is,  therefore,  regarded  as  some 
power  of  10,  and  the  exponent  of  that  power  is  the  logarithm  of 
the  number. 

2.  If  a number  is  an  exact  power  of  10,  its  logarithm  is  a 
whole  number . If  a number  is  not  an  exact  power  of  10,  its 
logarithm  is  composed  of  two  parts,  a ivhole  number  called  the 
Characteristic,  and  a decimal  part  called  the  Mantissa. 
Thus,  225  being  greater  than  102  and  less  than  103,  its  logarithm 
is  found  to  be  2.352183,  of  which  2 is  the  characteristic  and 
.352183  is  the  mantissa. 

3.  In  a table  of  logarithms,  the  mantissas  only  are  necessarily 
given.  The  characteristic  of  the  logarithm  of  a number  is  deter- 
mined by  one  of  the  two  following  rules : 

Rule  I. — The  characteristic  of  the  logarithm  of  any 
whole  number  is  positive,  and  numerically  1 less  than  the 
number  of  places  of  figures  in  the  given  number . 


LO 


ELEMENTS  OF  SURY EYING. 


[BOOK  I. 


Rule  II. — The  characteristic  of  the  logarithm  of  a 
decimal  fraction  is  negative , and  numerically  1 greater 
than  the  number  of  0’s  that  immediately  follow  the 
decimal  point. 

Note  1.  — In  the  logarithm  of  a decimal  fraction,  the  charac- 
teristic alone  is  negative,  the  mantissa  being  always  positive . 
This  fact  is  indicated  by  writing  the  negative  sign  over  the 
characteristic:  thus, 2.371465,  is  equivalent  to  — 2 + .371465. 

Note  2. — It  is  to  be  observed,  that  the  characteristic  of  the 
logarithm  of  a mixed  number  is  the  same  as  that  of  its  entire 
part.  Thus,  the  characteristic  of  the  logarithm  of  725.4275  is 
the  same  as  the  characteristic  of  the  logarithm  of  725. 

4.  A Table  of  Logarithms  is  a table  by  means  of  which 
may  be  found  the  logarithm  corresponding  to  any  number,  or  the 
number  corresponding  to  any  logarithm. 

In  the  table  appended,  the  mantissas  alone  are  given;  the 
characteristic  may  be  found  by  one  of  the  rules  of  Art.  3. 

The  mantissa  of  the  logarithm  of  any  number  is  not  changed 
by  multiplying  or  dividing  the  number  by  any  exact  power  of  10. 
Hence,  in  finding  the  mantissa  of  the  logarithm  of  a number,  the 
position  of  the  decimal  point  may  be  changed  at  pleasure.  Thus, 
the  mantissa  of  the  logarithm  of  456357,  is  the  same  as  that  of 
the  number  4563.57  ; and  the  mantissa  of  the  logarithm  of  75,  is 
the  same  as  that  of  7500. 

5.  To  find  the  logarithm  of  a number  between  1000  and 
10,000. — Find  the  characteristic  by  the  first  rule  of  Art.  3. 
To  determine  the  mantissa,  find  in  the  column  headed  “N”  the 
left  hand  three  figures  of  the  given  number;  then  pass  along  the 
horizontal  line  in  which  these  figures  are  found,  to  the  column 
headed  by  the  fourth  figure  of  the  given  number,  and  take  out 
the  four  figures  found  there ; pass  back  again  to  the  column 


SEC.  I.J 


LOGARITHMS. 


11 


headed  “0,”  and  there  will  be  found  in  this  column,  either  upon 
the  horizontal  line  of  the  first  three  figures  or  a few  lines  above 
it,  a number  consisting  of  six  figures,  the  left-hand  two  figures  of 
which  must  be  prefixed  to  the  four  already  taken  out.  Thus, 

Log  8979  = 3.953228. 

If,  however,  any  dots  are  found  at  the  place  of  the  four  figures 
first  taken  out,  or  if  in  returning  to  the  “ 0 ” column  any  dots 
are  passed,  the  two  figures  to  be  prefixed  are  the  left-hand  two  of 
the  six  figures  of  the  “ 0 ” column  immediately  below.  Dots  in 
the  number  taken  out  must  be  replaced  by  zeros.  Thus, 

Log  3098  = 3.491081 

Log  2188  = 3.340047 

6.  To  find  the  logarithm  of  a number  between  1 and  1000. — 

Find  the  characteristic  by  the  first  rule  of  Art.  3.  To  find  the 
mantissa,  fill  out  the  given  number  to  four  places  of  figures  (or 
conceive  it  to  be  so  filled  out)  by  annexing  0’s  (see  Art.  4),  and 
find  the  mantissa  corresponding  to  the  resulting  number,  as  in 
Art.  5.  Thus,  to  find  log.  of  75  : characteristic  is  1,  by  the  rule; 
the  mantissa  is  the  same  as  that  corresponding  to  7500,  i.  e,9 
.875061  ; hence, 

Log  75  = 1.875061. 

In  the  same  way, 

Log  2 = 0.301030. 

7.  To  find  the  logarithm  of  a number  greater  than  10,000. — 

Find  the  characteristic  by  the  first  rule  of  Art.  3.  To  find  the 
mantissa  : set  aside  all  of  the  given  number  except  the  left-hand 
four  figures,  and  find  the  mantissa  corresponding  to  these  four, 
as  in  Art.  5 ; multiply  the  corresponding  tabular  difference , 
found  in  column  “D,”  by  the  part  of  the  number  set  aside,  and 
discard  as  many  of  the  right-hand  figures  of  the  product  as  there 
are  figures  in  the  multiplier,  and  add  the  result  thus  obtained  to 


12 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


the  mantissa  already  found.  If  the  left-hand  figure  of  those  dis- 
carded is  5,  or  more,  increase  the  number  added  by  1. 

Note. — It  is  to  be  observed  that  the  tabular  difference,  found 
in  column  “D,”is  millionths,  and  not  a whole  number;  and 
that,  therefore,  the  result  to  be  added  “to  the  mantissa  already 
found  ” is  millionths. 

Example. — To  find  the  logarithm  of  672887:  the  character- 
istic is  5 ; set  aside  87,  and  the  mantissa  corresponding  to  6728  is 
.827886  ; the  corresponding  tabular  difference  is  65,  which  mul- 
tiplied by  87,  the  part  of  the  number  set  aside,  gives  5655  ; as 
there  are  two  figures  in  the  multiplier,  discard  the  right-hand 
two  figures  of  this  product,  leaving  56  ; but  as  the  left-hand 
figure  of  those  discarded  is  5,  call  the  result  57  (which  is 
millionths ) ; adding  this  57  to  the  mantissa  already  found,  will 
give  .827943  for  the  required  mantissa  ; hence, 

Log  672887  = 5.827943. 

In  the  same  way. 

Log  3710053  = 6.569380. 

8.  To  find  the  logarithm  of  a decimal. — Find  the  character- 
istic by  the  second  rule  of  Art.  3.  To  find  the  mantissa,  drop 
the  decimal  point  and  consider  the  decimal  a whole  number. 
Find  the  mantissa  of  the  logarithm  of  this  number  as  in  preced- 
ing articles,  and  it  will  be  the  mantissa  required.  Thus, 

Log  .0327  = 2.514548 
Log  .378024  = T.  577520. 

Note. — To  find  the  logarithm  of  a mixed  number , find  the 
characteristic  by  Note  2,  Art.  3 ; then  drop  the  decimal  point 
and  proceed  as  above. 

9.  To  find  the  number  corresponding  to  a given  logarithm. — 

The  rule  is  the  reverse  of  those  just  given.  Look  in  the  table  for 
the  mantissa  of  the  given  logarithm.  If  it  cannot  be  found,  take 


SEC.  I.] 


LOGARITHMS. 


13 


out  the  next  less  mantissa,  and  also  the  corresponding  number, 
which  set  aside.  Find  the  difference  between  the  mantissa  taken 
out  and  that  of  the  given  logarithm  ; annex  any  number  of  0’s, 
and  divide  this  result  by  the  corresponding  number  in  the  column 
44  D.”  Annex  the  quotient  to  the  number  set  aside,  and  then,  if 
the  characteristic  is  positive,  point  off,  from  the  left  hand,  a num- 
ber of  places  of  figures  equal  to  the  characteristic  plus  1 ; the 
result  will  be  the  number  required. 

If  the  characteristic  is  negative,  prefix  to  the  figures  obtained 
a number  of  0’s  one  less  than  the  number  of  units  in  the  nega- 
tive characteristic  and  to  the  whole  prefix  a decimal  point ; the 
result,  a pure  decimal,  will  be  the  number  required. 

Example. — Let  it  be  required  to  find  the  number  correspond- 
ing to  the  logarithm  5.233568. 

The  next  less  mantissa  in  the  table  is  233504  ; the  correspond- 
ing number  is  1712,  and  the  tabular  difference  is  253. 

OPERATION  . 

Given  mantissa, 233568 

Next  less  mantissa,  ....  233504  . . 1712 

253  ) 6400000  ( 25296 

.-.  The  required  number  is  171225.296. 

The  number  corresponding  to  the  logarithm  2.233568  is 
.0171225  + . 

10.  Multiplication  by  Logarithms. — Rule. — Find  the 
logarithms  of  the  factors  and  take  their  sum;  then  find 
the  number  corresponding  to  the  resulting  logarithm,  and 
it  will  be  the  product  required. 

Example. — Find  the  continued  product  of  3.902,  5971.6, 
and  .0314728. 


14 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


OPERATION. 

log  3.902  = 0.591287 
log  5971.6  = 3.776091 
log  .0314728  — 2.497936 

2.865314  73o.  54,  product 

Here,  the  1 carried  added  to  the  3 gives  4,  which  added  to  —2 
gives  2 as  the  characteristic  of  the  logarithm  of  the  product. 

11.  Division  by  Logarithms. — Rule. — Find  the  loga- 
rithms of  the  dividend  and  divisor,  and  subtract  the  latter 
from  the  former ; then  find  the  number  corresponding  to 
the  resulting  logarithm,  and  it  will  be  the  quotient  re- 
quired. 

EXAMPLES. 

1.  Divide  24163  by  4567. 

OPERATION. 

log  24163  . . . 4.383151 
log  4567  . . . 3.659631 

0.723520  5.29078,  quotient. 


2.  Divide  0.7438  by  12.9476. 

OPERATION. 

log  0.7438.  . . 1.871456 

log  12.9476  . . . 1.112189 

2.759267  . *.  0.057447,  quotient. 

Here,  1 taken  from  T,  gives  2 for  a result.  The  subtraction, 
as  in  this  case,  is  always  to  be  performed  in  the  algebraic  sense. 

The  operation  of  division,  particularly  when  combined  with 
that  of  multiplication,  can  often  bo  simplified  by  using  the 
principle  of  the 


SEC.  I.] 


LOGARITHMS. 


15 


12.  Arithmetical  Complement. — The  arithmetical  comple- 
ment of  a logarithm  is  the  remainder  obtained  by  subtracting  it 
from  10.  Thus,  8.130456  is  the  arithmetical  complement  of 
1.869544.  Thr  arithmetical  complement  is  denoted  by  the 
symbol  (a.  c.j^- 

The  following  is  the  rule  for  the  use  of  the  arithmetical  com- 
plement in  division  by  logarithms : 

Kule. — Find  the  logarithm  of  the  dividend,  and  the 
arithmetical  complement  of  the  logarithm  of  the  divisor, 
add  them  together,  and  diminish  the  sum  by  10;  the 
number  corresponding  to  the  resulting  logarithm  will  be 
the  quotient  required. 

Examples. — 1.  Divide  37.149  by  523.76. 
log  37.149  . . . 1.569947 

(a.  c.)  log  523.76  . . . 7.280867 

2.850814  .*.  0.0709273,  quotient. 

The  operation  of  subtracting  10  is  performed  mentally. 

2.  Find  x in  the  proportion, 

602.647  : 2.29863  ::  x : .037293. 
log  602.647  . . 2.780063 

(a.  c.)  log  2.29863  . . 9.638531 

log  .037293  . . 2.571627 

log  z ...  . 0.990221  .;.  z = 9.7773  + . 

3.  Divide  the  product  of  3.58884  and  5672,  by  the  product  of 
89721  and  42.056. 

log  358884  . . . 5.554954 

log  5672  ....  3.753736 

(a.  c.)  log  89721  . . . 5.047106 

(a.  c.)  log  42.056  . . . 8.376182 

2.731978  . • . 539.48,  result. 

20  is  here  subtracted,  as  (a.  c.)  has  been  twice  used. 


16 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


Note.— If  the  logarithm,  whose  arithmetical  complement  is 
taken,  exceeds  10,  subtract  it  from  20,  and  reject  20  in  the  final 
operation. 

13.  Raising  to  Powers  by  Logarithms.—  Rule.— Find 
the  logarithm  of  the  number,  and  multiply  it  by  the 
exponent  of  the  power ; then  find  the  number  correspond- 
ing to  the  resulting  logarithm,  and  it  will  be  the  power 
required. 

Example. — Find  the  5th  power  of  9. 


log  9 0.954243 

5 


4.771215  .*.  59049,  power. 


14.  Extracting  Roots  by  Logarithms.— UuiM.—Find 
the  logarithm  of  the  number  and  divide  it  by  the  index 
of  the  root;  then  find  the  number  corresponding  to  the 
resulting  logarithm,  and  it  will  be  the  root  required. 

Example. — Find  the  cube  root  of  4096. 

The  logarithm  of  4096  is  3.612360,  and  one-third  of  this  is 
1.204120.  The  corresponding  number  is  16,  which  is  the  root 
sought. 

If  the  characteristic  of  the  logarithm  of  the  given  number  is 
negative  and  not  exactly  divisible  by  the  index  of  the  root,  add 
to  it  such  negative  quantity  as  shall  make  it  exactly  divisible,  and 
add  also  to  the  mantissa  a numerically  equal  positive  quantity. 

Thus,  let  the  square  root  of  .00863  be  required. 


log  .00863  = 3.936011  = 4 + 1.936011. 


leg 


The  number  sought  is  therefore  .09289  + . 


SEC.  II.  | 


PLANE  TRIGONOMETRY. 


1? 


SECTION  II. 

PLANE  TRIGONOMETRY. 

15.  Plane  Trigonometry  is  that  branch  of  Mathematics 
which  treats  of  the  solution  of  plane  triangles. 

In  every  plane  triangle  there  are  six  parts  : three  sides  and 
three  angles.  When  three  of  these  parts  are  given,  one  being  a 
side,  the  remaining  parts  may  be  found  by  computation.  The 
operation  of  finding  the  unknown  parts,  is  called  the  solution 
of  the  triangle. 

16.  A plane  angle  is  measured  by  the  arc  of  a circle  included 
between  its  sides,  the  centre  of  the  circle  being  at  the  vertex, 
and  its  radius  being  equal  to  1. 

Thus,  if  the  vertex  A be  taken  as  a 
centre,  and  the  radius  A B be  equal  to  1, 
the  intercepted  arc  BC  will  measure  the 
angle  A. 

Let  A BCD  represent  a circle  whose  radius  is  equal  to  1, 
and  A (7,  BD , two  diameters  perpendicu- 
lar to  each  other.  These  diameters  divide 
the  circumference  into  four  equal  parts, 
called  quadrants;  and  because  each  of 
the  angles  at  the  centre  is  a right  angle, 
it  follows  that  a right  angle  is  measured 
by  a quadrant.  An  acute  angle  is  meas-  ^ 

ured  by  an  arc  less  than  a quadrant , 
and  an  obtuse  angle , by  an  arc  greater  than  a quadrant. 

17.  In  Geometry,  the  unit  of  angular  measure  is  a right 
angle ; so  in  Trigonometry,  the  'primary  unit  is  a quadrant , 
which  is  the  measure  of  a right  angle. 


18 


ELEMENTS  OF  SURV  KYI  NO. 


[BOOK  1. 


For  convenience,  the  quadrant  is  divided  into  90  equal  parts, 
each  of  which  is  called  a degree  ; each  degree  into  00  equal  parts, 
called  minutes;  and  each  minute  into  60  equal  parts,  called 
seconds.  Degrees,  minutes,  and  seconds,  are  denoted  by  the 
symbols  °,  ',  Thus,  the  expression  7°  22'  33",  is  read, 
7 degrees,  22  minutes , and  33  seconds.  Fractional  parts  of  a 
second  are  expressed  decimally. 

18.  The  complement  of  an  arc  is. the  difference  between  that 
arc  and  90°.  The  complement  of  an 
angle  is  the  difference  between  that 
angle  and  a right  angle. 

Thus,  EB  is  the  complement  of 
AE,  and  FB  is  the  complement  of 
OF.  In  like  manner,  EOB  is  the 
complement  of  A OE,  and  FOB  is 
the  complement  of  COF. 

In  a right-angled  triangle,  the  acute  angles  are  complements 
of  each  other. 

19.  The  supplement  of  an  arc  is  the  difference  between  that 
arc  and  180°.  The  supplement  of  an  angle  is  the  difference 
between  that  angle  and  two  right  angles. 

Thus,  EC  (Fig.  3)  is  the  supplement  of  AE,  and  FC  the 
supplement  of  AF.  In  like  manner,  EOC  is  the  supplement 
of  AOE,  and  FOC  the  supplement  of  AOF. 

In  any  plane  triangle,  either  angle  is  the  supplement  of  the 
sum  of  the  other  two. 

20.  Instead  of  the  arcs  themselves,  certain  functions  of  the 
arcs,  as  explained  below,  are  used.  A function  of  a quantity  is 
something  which  depends  upon  that  quantity  for  its  value. 

The  following  functions  are  the  only  ones  needed  for  solving 
triangles : 


B 


1 0 

Fig  3 


SEC.  II.] 


PLANE  TRIGONOMETRY. 


19 


21.  The  sine  of  an  arc  is  the  distance  of  one  extremity  of 
the  arc  from  the  diameter  through  the  other  extremity. 

Thus,  PM  (Fig.  4)  is  the  sine 
of  AM,  and  P'M'  is  the  sine  of 
AM'. 

22.  The  cosine  of  an  arc  is 
the  sine  of  the  complement  of 
the  arc,  “complement  sine”  be- 
ing contracted  into  cosine. 

Thus,  NM  (Fig.  4)  is  the  co- 
sine of  AM,  and  NM'  is  the 
cosine  of  AM'.  These  lines  are  respectively  equal  to  OP 
and  OP'. 

23.  The  tangent  of  an  arc  is  the  perpendicular  to  the  radius 
at  one  extremity  of  the  arc,  limited  by  the  prolongation  of  the 
diameter  through  the  other  extremity. 

Thus,  AT  (Fig.  4)  is  the  tangent  of  the  arc  AM,  and  AT"' 
is  the  tangent  of  the  arc  AM'. 

24.  The  cotayigent  of  an  arc  is  the  tangent  of  its  complement, 
“complement  tangent”  being  contracted  into  cotangent. 

Thus,  BT  (Fig.  4)  is  the  cotangent  of  the  arc  AM,  and 
BT"  is  the  cotangent  of  the  arc  AM'. 

The  sine,  cosine,  tangent,  and  cotangent  of  an  arc,  a,  are, 
for  convenience,  written  sin  a,  cos  a,  tan  a,  and  cot  a. 

These  functions  of  an  arc  may  also  be  considered  as  func- 
tions of  the  angle  which  the  arc  measures. 

Thus,  (Fig.  4)  PM,  NM,  AT,  and  BT',  are  respectively  the 
sine,  cosine,  tangent,  and  cotangent  of  the  angle  A OM,  as  well 
as  of  the  arc  AM. 

25.  The  sine  of  an  arc  is  equal  to  the  sine  of  its  supplement ; 
and,  in  general,  any  function  of  an  arc  is  equal  to  the  corres- 


20 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


ponding  function  of  its  supplement.  Thus,  if  A is  any  arc 
or  angle, 

sin  A = sin  (180°  — A)  ; 
cos  A = cos  (180°  — A) ; 
tan  A = tan  (180°  — A) ; 
cot  A = cot  (180°  — A). 


Note. — These  relations  exist  between  the  numerical  values 
of  the  functions ; the  algebraic  signs , which  they  have  in  the 
different  quadrants,  are  not  considered. 


26.  A Natural  Sine,  Cosine,  Tangent,  or  Cotangent, 

is  the  sine,  cosine,  tangent,  or  cotangent,  of  an  arc  whose 
radius  is  1. 


A Table  of  Natural  Sines  is  a table  from  which  the 
natural  sine,  cosine,  tangent,  or  cotangent  of  any  arc  may 
be  found. 

The  Table  of  Natural  Sines,  beginning  at  page  63  of  the 
tables,  gives  the  values  of  the  sines  and  cosines  only.  If  the 
tangent  or  cotangent  of  an  arc.  A,  is  desired,  it  may  be  found 
by  the  relation, 


tan  A — 


sin  A 
cos  A 


cot  A = 


cos  A 
sin  A 


TABLE  OF  LOGARITHMIC  SINES. 

27.  A Logarithmic  Sine,  Cosine,  Tangent,  or  Cotan- 
gent is  the  logarithm  of  the  sine,  cosine,  tangent,  or  cotangent 
of  an  arc  whose  radius  is  10,000,000,000. 

A Table  of  Logarithmic  Sines  and  Tangents  is  a table 
giving  the  logarithm  of  the  sine  and  cosine,  tangent  and  cotan- 
gent of  any  arc  or  angle. 

The  logarithm  of  the  tabular  radius  is  10. 


SEC.  II.] 


PLANE  TRIGONOMETRY. 


21 


Any  logarithmic  function  of  an  arc  or  angle  may  be  found  by 
multiplying  the  corresponding  natural  function  by  10,000,000,000, 
and  then  taking  the  logarithm  of  the  result ; or  more  simply,  by 
taking  the  logarithm  of  the  corresponding  natural  function,  and 
then  adding  10  to  the  result. 

28.  In  the  table,  beginning  at  page  18  of  the  tables,  the 
logarithmic  functions  are  given  for  every  minute  from  0°  to  90°. 
In  addition,  their  rates  of  change  for  each  second , are  given  in 
the  column  headed  “ D.” 

For  the  sine  and  cosine,  there  are  separate  columns  of  dif- 
ferences, which  are  written  to  the  right  of  the  respective 
columns;  but  for  the  tangent  and  cotangent,  there  is  but 
a single  column  of  differences,  which  is  written  between 
them. 

The  angle  obtained  by  taking  the  degrees  from  the  top  of  the 
page,  and  the  minutes  from  any  line  on  the  left  hand  of  the 
page,  is  the  complement  of  that  obtained  by  taking  the  degrees 
from  the  bottom  of  the  page,  and  the  minutes  from  the  same  line 
on  the  right  hand  of  the  page.  But,  by  definition,  the  cosine  and 
the  cotangent  of  an  arc  are,  respectively,  the  sine  and  the 
tangent  of  the  complement  of  that  arc  (Arts.  22  and  24)  ; 
hence,  the  columns  designated  sine  and  tang,  at  the  top  of  the 
page,  are  designated  cosine  and  cotang  at  the  bottom. 

29.  To  find  the  logarithmic  functions  of  an  angle  which  is 
expressed  in  degrees  and  minutes. 

If  the  angle  is  less  than  45°,  look  for  the  degrees  at  the  top 
of  the  page,  and  the  minutes  in  the  left  hand  column ; then 
follow  the  corresponding  horizontal  line  to  the  column  desig- 
nated at  the  top  by  sine,  cosine,  tang,  or  cotang,  as  the  case 
may  be ; the  number  there  found  is  the  logarithm  required. 
Thus, 


22 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


log  sin  19°  55'  . . . 9.532312 

log  tan  19°  55'  . . . 9.559097 

If  the  angle  is  greater  than  45°,  look  for  the  degrees  at  the 
bottom  of  the  page,  and  for  the  minutes  in  the  right  hand 
column  ; then  follow  the  corresponding  horizontal  line  back- 
wards to  the  coin mn  designated  at  the  bottom  by  sine,  cosine, 
tang,  or  cotang,  as  the  case  may  be ; the  number  there  found  is 
the  logarithm  required.  Thus, 

log  cos  52°  18'  . . . 9.780416 

log  tan  52°  18'  . . . 10.111884 

30.  To  find  the  logarithmic  functions  of  an  angle  which  is 
expressed  in  degrees,  minutes,  and  seconds. 

Find  the  logarithm  corresponding  to  the  degrees  and  minutes 
as  before  ; then  multiply  the  corresponding  number  taken  from 
the  column  headed  “ D ” (which  is  millionths),  by  the  number  of 
seconds,  and  add  the  product  to  the  preceding  result,  for  the 
sine  or  tangent,  and  subtract  it  therefrom  for  the  cosine  or 
cotangent. 

EXAMPLES. 

1.  Find  the  logarithmic  sine  of  40°  26'  26". 


OPERATION. 

log  sin  40°  26' 9.811952 

Tabular  difference  2.47 
No.  of  seconds  . 28 

Product.  . . . 69.16  to  be  added  . 69 

log  sin  40°  26'  28" 9.812021 


The  same  rule  is  followed  for  the  figures  discarded  (in  this 
case  16),  as  in  Art.  7. 


SEC.  II. J 


PLANE  TRIGONOMETRY. 


23 


% Find  the  logarithmic  cosine  of  53°  40'  46". 


OPERATION. 

log  cos  53°  40' 9.772675 

Tabular  difference  2.86 
No.  of  seconds  . 46 

Product  . . . 131.56  to  be  subtracted  132 

log  cos  53°  40'  46" 9.772543 


If  the  angle  is  greater  than  90°,  we  find  the  required  function 
of  its  supplement  (Art.  25). 

3.  Find  the  logarithmic  tangent  of  118°  18'  25". 


OPERATION. 

180° 

Given  arc 118°  18'  25" 

Supplement 61°  41'  35" 

log  tan  61°  41' 10.268556 

Tabular  difference  5.04 
No.  of  seconds  . 35 

Product  . . .176.40  to  be  added  . 176 

log  tan  118°  18'  25" 10.268732 


31.  To  find  the  angle  corresponding  to  any  logarithmic 

function. 

This  is  done  by  reversing  the  preceding  rule:  Look  in  the 
proper  column  of  the  table  for  the  given  logarithm ; if  it  is 
found  there,  the  degrees  are  to  be  taken  from  the  top  or  bottom, 
and  the  minutes  from  the  left  or  right  hand  column,  as  the  case 
may  be.  If  the  given  logarithm  is  not  found  in  the  table,  then 
find  the  next  less  logarithm,  and  take  frorp  the  table  the  corres- 
ponding degrees  and  minutes,  and  set  them  aside.  Subtract  the 
logarithm  found  in  the  table,  from  the  given  logarithm,  annex 


24 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


two  0’s  to  the  remainder,  and  divide  this  result  by  the  corres- 
ponding tabular  difference.  The  quotient  will  be  seconds,  which 
must  be  added  to  the  degrees  and  minutes  set  aside,  in  the  case 
of  a sine  or  tangent,  and  subtracted , in  the  case  of  a cosine  or  a 
cotangent. 

examples. 

1.  Find  the  angle  corresponding  to  the  logarithmic  sine 
9.422248. 

OPERATION. 

Given  logarithm  . . . 9.422248 

Next  less  in  table  . . . 9.421857  ...  15°  19' 

Tabular  difference  7.68  ) 391.00  ( 51",  to  be  added. 

Hence,  the  required  arc  is  15°  19'  51". 

2.  Find  the  angle  corresponding  to  the  logarithmic  cosine 
9.427485. 

OPERATION. 

Given  logarithm  . . . 9.427485 

Next  less  in  table  . . . 9.427354  . . . 74°  29' 

Tabular  difference  7.58  ) 131.00  ( 17",  to  be  subt. 

Hence,  the  required  angle  is  74°  28'  43". 


32.  Theorem  I. — The  sides  of  a plane  triangle  are  propor- 
tional to  the  sines  of  their  opposite  angles. 


In  the  triangle  ABC 
let  the  large  letters  A,  B , 
C\  designate  the  angles, 
and  the  corresponding 
small  letters,  a,  b,  c,  the 
sides  opposite  ; then, 

a : b 
a : c 
b : c 


: sin  A : sin  B\ 
: sin  A : sin  C\ 
: sin  B : sin  C. 


Fig.  5. 


SEC.  II.  J 


PLANE  TRIGONOMETRY. 


25 


33.  Theorem  II. — In  any  plane  triangle , the  sum  of  the 
two  sides  containing  any  angle , is  to  their  difference , as  the 
tangent  of  half  the  sum  of  the  two  other  angles  is  to  the  tangent 
of  half  their  difference. 

Tims,  in  the  triangle  ABC  (Fig.  5), 

a + b : a — b ::  tan  \{A-\-B)  : tan  J (.T  — B) ; 

a-\-c  : a — c : : tan  ^ {A  -f-  C)  : tan  J (A  — C)  ; 

b + c : b — c : : tan  \ (B-\-  C)  : tan  \ (B—  C). 

By  solving  any  one  of  the  above  proportions,  the  first  three 

terms  being  known,  the  tangent  of  hall*  the  difference  of  the  two 
unknown  angles  is  obtained,  and  from  this  tangent  the  half 
difference  itself  is  found.  The  greater  of  the  two  unknown 
angles  is  equal  to  half  their  sum  added  to  half  their  difference: 
the  smaller  is  equal  to  half  their  sum  diminished  by  half 
their  difference. 

34.  Theorem  III. — In  any  plane  triangle , if  a line  is 
drawn  from  the  vertex  of  the  vertical  angle  perpendicular  to 
the  base , dividing  it  into  two  segments : then , the  sum  of  the  two 
segments,  or  the  whole  base,  is  to  the  sum 
of  the  two  other  sides,  as  the  difference  of 
those  sides,  to  the  difference  of  the  segments. 

Thus,  in  the  triangle  ABC  (Fig.  6), 
s + s'  : b-fc  ::  b—c  : s—s\ 

35.  Theorem  IV.  — In  any  right-angled  plane  triangle, 
radius  is  to  the  tangent  of  either  of  the  acute  angles,  as  the  side 
adjacent  to  the  side  opposite. 

Let  CAB  (Fig.  7)  be  the  proposed  triangle,  and  denote  the 
radius  by  R : then  will 


26 


ELEMENTS  OF  SURVEYING 


[BOOK  ]. 


R*  : tan  C ; : AC,  or  b : AB,  or  c ; 

Also, 

R*  : tan  B : : AB,  or  c : AC,  or  b. 

36.  Theorem  V. — In  any  right-angled  triangle,  radius  is 
to  the  cosine  of  either  of  the  acute  angles,  as  the  hypotlienuse  to 
the  side  adjacent. 

In  the  triangle  CAB  (Fig.  7), 

R*  : cos  C : : BC,  or  a : AC,  or  b ; 

Also, 

R*  : cos  B : : BC,  or  a : AB,  or  c. 

37.  Solution  of  Triangles. — The  relations  between  the 
sides  and  angles  of  plane  triangles,  stated  in  these  five  theorems, 
are  sufficient  to  solve  all  the  cases  of  Plane  Trigonometry.  Of 
the  six  parts,  which  make  up  a plane  triangle,  three  must  be 
given,  and  at  least  one  of  these  must  be  a side,  before  the  others 
can  be  determined. 

If  the  three  angles  only  are  given,  it  is  plain  that  an  indefinite 
number  of  similar  triangles  may  be  constructed,  the  angles  of 
which  may  be  respectively  equal  to  the  angles  that  are  given 
and,  therefore,  the  sides  could  not  be  determined. 

Assuming,  with  this  restriction,  any  three  parts  of  a triangle, 
one  of  the  four  following  cases  will  always  be  presented: 

I.  When  two  angles  and  a side  are  given. 

II.  When  two  sides  and  an  angle  opposite  one  of  them 
are  given. 

IIT.  When  two  sides  and  the  included  angle  are  given. 

IV.  When  the  three  sides  are  given. 

* If  logarithmic  functions  are  uncd,  II  in  equal  to  10,000,000,000,  and  its  logarithm  is 
10;  otherwise,  Jl  is  equal  to  1. 


Fio.  7. 


SEC.  II. j 


PLANE  TRIGONOMETRY. 


27 


CASE  I. 

38.  When  two  angles  and  a side  are  given. 

In  a plane  triangle,  ABC \ there  are  given  the  angle 
A = 58°  07',  the  angle  B = 22°  37',  and  the  side  AB,  or 
c — 408  yards ; to  find  C,  a,  and  b.  (The  sides  lying  opposite 
the  angles  A , B,  and  C,  are  denoted  by  a,  b , and  c. 

Add  the  given  angles,  A and  B,  to- 
gether, and  subtract  their  sum  from 
180° ; the  remainder  will  he  the  other 
angle,  C.  Then  from  the  proportion  Fig.  a 

(Theorem  I), 

sin  C : sin  A : : c : a ; a may  be  found  ; 
and  from  the  proportion, 

sin  C : sin  B : : c : b ; b may  be  found. 


CASE  II. 

39.  When  two  sides  and  an  angle  opposite  one  of  them  are 

given. 

In  a plane  triangle,  ABC,  there  are  given  AC,  or  b = 216, 
CB,  or  a = 117,  the  angle  A = 22°  37',  to  find  the  other  parts. 


GEOMETRICALLY 


Draw  an  indefinite  right  line  AB  B ; 
from  any  point,  as  A,  draw  AC,  making 
BA C = 22°  37'  and  make  AC  = 216. 

With  C as  a centre,  and  a radius  equal 
to  117,  the  other  given  side,  describe  the 

arc  B'B;  draw  CB  and  CB' ; then  will  either  of  the  triangles, 
ACB  or  ACB',  answer  all  the  conditions  of  the  question. 


28 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


TRIGONOMETRICALLY. 

From  Theorem  I,  we  have, 

a : b : : sin  A : sin  B. 

By  applying  logarithms,  we  have, 

(a.  c.)  log  a (117) 7.931814 


log  b (21G)  . . 

log  sin  A (22°  37') 


2.334454 

9.584908 


log  sin  B,  45°  13'  55",  or  134°  46'  05" 


9.851230 


The  ambiguity  in  this  and  similar  examples,  arises  in  conse- 
quence of  the  first  proportion  being  true  for  either  of  the  angles 
ABC,  or  AB'C,  which  are  supplements  of  each  other  and,  there- 
fore, have  the  same  sine  (Art.  25).  So  long  as  the  two  triangles 
ACB  and  ACB'  exist,  the  ambiguity  will  continue.  But  if  the 
side  CB,  opposite  the  given  angle,  is  greater  than  AC,  the  arc 
BB' , described  from  C as  a centre  and  with  a radius  equal  to  the 
side  a,  will  cut  the  line  ABB',  on  the  same  side  of  the  point  A, 
in  but  one  point,  and  then  there  will  be  only  one  triangle 
answering  to  the  conditions. 

If  the  side  CB  is  equal  to  the  perpendicular  Cd,  the  arc  BB' 
will  be  tangent  to  ABB' , and  in  this  case  also,  there  will  be  but 
one  triangle.  When  CB  is  less  than  the  perpendicular  Cd,  the 
arc  BB'  will  not  intersect  the  base  ABB' , and  in  that  case  no 
triangle  can  be  formed,  or  it  will  be  impossible  to  fulfill  the  con- 
ditions of  the  problem. 

In  the  example  under  consideration,  there  are  two  solutions, 
the  first  corresponding  to  B — 45°  13'  55",  and  the  second  to 
AB'C  = 134°  40'  05". 


FIRST  CASE. 


180-07°  50'  55"  = 112°  09'  05". 


A . . 

B . . 

C . . 


SEC.  II.] 


PLANE  TRIGONOMETRY. 


29 


Then,  in  the  triangle  A CB , 

sin  B : sin  C : : b : c, 
and  applying  logarithms. 


(a.  c)  log  sin  B (45°  13'  55")  ....  0.148764 

log  sin  C (412°  09'  05")  ....  9.966700 

log  b (216) 2.334454 

log  c 281.785  2.449918 


SECOND  CASE. 


A 22°  37' 

B 134°  46'  05" 

C' 180°  — 157°  23'  05"  = 22°  36'  55". 


Then,  in  the  triangle  AC'B', 

sin  B'  : sin  C'  : b : c’9 
and  applying  logarithms, 

(a.  c.)  log  sin  B (134°  46'  05")  ....  0.148764 
log  sin  C'  (22°  36'  55")  ....  9.584943 


log  b (216) 2.334454 

log  c'  116.993  2.068161 


2.  Given  two  sides  of  a triangle,  50  and  40  respectively,  and 
the  angle  opposite  the  latter,  equal  to  32° ; required  the  remaining 
parts  of  the  triangle. 

Ans.  If  the  angle  opposite  the  side  50  is  acute,  it  is  equal  to 
41°  28'  59";  the  third  angle  is  then  equal  to  106°  31'  01",  and 
the  third  side  to  72.368.  If  the  angle  opposite  the  side  50  is 
obtuse,  it  is  equal  to  138°  31'  01",  the  third  angle  to  9°  28'  59", 
and  the  remaining  side  to  12.436. 


30 


ELEMENTS  OF  SURVEYING. 


[BOOK  1. 


CASE  III. 

40.  When  two  sides  and  their  included  angle  are  given. 

Let  ABC  be  a triangle  ; AB  and  BC \ a 

the  given  sides,  and  B the  given  angle. 

Since  B is  known,  we  can  find  the 
sum  of  the  two  other  angles ; for, 

Fig.  10. 

A + C = 180  °~B, 

and 

+ O)  = * (180 °-B). 

We  next  find  half  the  difference  of  the  angles  A and  (7,  by 
Theorem  II,  viz., 


BC-\-BA  : BC—BA  ::  tan  \(A  + C)  : tan  | (A  — C), 

in  which  we  consider  BC  greater  than  BA , and  therefore  A is 
greater  than  C ; since  the  greater  angle  must  be  opposite  the 
greater  side. 

Having  found  half  the  difference  of  A and  C,  by  adding  it  to 
the  half  sum,  \ (A  + C),  we  obtain  the  greater  angle,  and  by 
subtracting  it  from  half  the  sum,  we  obtain  the  less.  That  is, 

l(A  + C)+±{A-C)=A, 

and 

t(A  + C)-\(A-C)  = C. 

Having  found  the  angles  A and  C,  the  third  side  AC  may  be 
found  by  the  proportion, 


sin  A : sin  B : : a : b. 


SEC.  II. J 


PLANE  TRIGONOMETRY. 


31 


CASE  IV. 

41.  Having  given  the  three  sides  of  a plane  triangle  to  find 

the  angles. 

Let  fall  a perpendicular  from  the  angle  opposite  the  greatest 
side,  dividing  the  given  triangle  into  two  right-angled  triangles  ; 
then  find  the  difference  of  the  segments  of  the  base  by  Theorem 
III.  Half  this  difference  being  added  to  half  the  base,  gives  the 
greater  segment ; and,  being  subtracted  from  half  the  base,  gives 
the  less  segment ; the  greater  segment  belongs  to  the  right- 
angled  triangle  having  the  greater  hypothenuse.  We  then 
have  two  sides  and  the  right  angle  of  each  of  two  right-angled 
triangles,  to  find  the  acute  angles. 


Example. — The  sides  of  a plane 
triangle  being  given ; viz.,  BC  — 40, 
AC  = 34,  and  AB  = 25  ; required  the 
angles. 


BC 

: AC+AB  :: 

AC-AB  : 

That  is, 

40  : 59  ::  9 

. 59  x 9 
‘ 40  “ 

Then, 

40  + 13.275 

2 

= 26.6375  = 

And, 

40  + 13.275 

2 

= 13.3625  = 

In  the  triangle  DAC,  to  find  the  angle  DAC. 

AC  : DC  ::  sin  D : sin  DAC. 


Applying  logarithms,  we  have, 

(a.  c.)  log  A C (34) 8.408521 

log  DC  (26.6375) 1.425493 

log  sin  D (90°) 10.000000 

log  sin  DAC  51°  34'  40"  . . . 9.894014 


32 


ELEMENTS  OF  SURVEYING. 


[BOOK  I. 


In  the  triangle  BAD,  to  find  the  angle  BAD. 

AB  : BD  ::  sin  D : sin  BAD. 


Applying  logarithms,  we  have, 

(a.  c.)  log  AB  (25) 8.602060 

log  BI)  (13.3625) 1.125887 

log  sin  D (90°) 10.000000 

log  sin  BAD  32°  18'  35"  . . . 9.727947 


Hence,  90°—/) A C = 90°  — 51°  34'  40"  = 38°  25'  20"  = C, 
and,  90°  — BAD  = 90°  — 32°  18'  35"  = 57°  41'  25"  = B, 

and,  BAD-\-DAC  = 51°  34'  40" + 32°  18'  35" 

= 83°  53'  15"  = A. 

42.  Solution  of  Right-angled  Triangles. — The  un- 
known parts  of  a right-angled  triangle  may  be  found  by  one  of 
the  last  four  cases  ; or,  if  two  of  the  sides  are  given,  by  means  of 
the  property  that  the  square  of  the  hypothenuse  is  equal  to  the 
sum  of  the  squares  of  the  two  other  sides, 
found  by  Theorems  IV.  and  V. 

EXAMPLES. 

1.  In  a right-angled  triangle  BAC, 
there  are  given  the  hypothenuse  BC  = 

250,  and  the  base  ^(7  = 240;  required 
the  other  parts. 

Ans.  B = 73°  44'  23";  C = 16°  15'  37";  AB  = 70. 

2.  In  a right-angled  triangle  BA  C,  there  are  given, 

AO  - 384,  and  B = 53°  08' ; 
required  the  remaining  parts. 

Ans.  A B — 287.96  ; BC  = 479.979  ; O = 36°  52'. 


SEC.  III.] 


INSTRUMENTS  FOR  PLOTTING. 


33 


SECTION  III. 

INSTRUMENTS  FOR  PLOTTING. 

43.  The  ordinary  implements  for  making  a diagram  or  plot 
of  a survey  are— drawing  board  ; T-square  ; dividers  ; ruler  and 
triangle  ; scale  of  equal  parts  ; semicircular  protractor. 


44.  A Drawing  Board  (Fig.  13)  is  a rectangular  board  of 
about  24  by  30  inches,  fths  of  an  inch  thick,  made  of  several 
pieces  of  well  seasoned  white  pine,  fitted  together  with  the  grain 
running  in  different  directions  to  prevent  warping.  It  is 
important  that  its  angles  should  be  perfect  right  angles. 


O 


Fig.  14. 

45.  A T-square  (Fig.  14)  is  a ruler  about  2 feet  in  length  let 
into  a thicker  piece  of  wood  at  right  angles  to  it.  One  side  of  the 
cross-piece  is  even  with  the  ruler  and  the  other  side  projects 
somewhat,  giving  a shoulder  on  that  side.  It  is  a convenient 
instrument  for  drawing  parallels  and  perpendiculars. 

3 


34 


ELEMENTS  OF  SURVEYING. 


[ROOK  I, 


46.  The  Dividers  (Fig.  15) 

consists  of  two  legs  ba , be,  which 
may  be  easily  turned  around  a 
joint  at  b. 

One  of  the  principal  uses  of 

this  instrument  is  to  lay  off  on  a line,  a distance  equal  to  a given 
line. 


Fig.  15. 


For  example,  to  lay  off  on  CD,  a distance  equal  to  AB. 

Open  the  points  of  the  dividers  to  a ai (R 

greater  distance  than  AB,  place  one  point 

lightly  upon  A,  and  then  slowly  and  con-  Cl~  e n 

tinuously  close  them  till  the  point  reaches 

B.  Then  raise  the  dividers,  place  one  foot  at  C,  and  mark  with 

the  other  the  distance  CE : this  will  evidently  be  equal  to  AB. 


Fig.  17. 

47.  Ruler  and  Triangle. — A Ruler  of  convenient  size  is 
about  twenty  inches  in  length,  two  inches  wide,  and  a fifth 
of  an  inch  in  thickness.  It  should  be  made  of  a hard  material, 
perfectly  straight  and  smooth. 

The  hypothenuse  of  the  right-angled  triangle,  which  is  used  in 
connection  with  it,  should  be  about  ten  inches  in  length,  and  it 
is  most  convenient  to  have  one  of  the  sides  considerably  longer 
than  the  other. 

The  two  following  problems  may  be  solved  with  the  ruler  and 
triangle. 


SEC.  III.] 


INSTRUMENTS  FOR  PLOTTING. 


35 


I.  To  draw  through  a given  point  a line  which  shall  be  par- 

allel to  a given  line. 

48.  Let  C be  the  given  point,  and  AB  the  given  line. 

Place  the  hypothenuse  of  the  triangle 

c 

against  the  edge  of  the  ruler,  and  then  — L— - 

place  the  ruler  and  triangle  on  the  paper,  A b 

so  that  one  of  the  sides  of  the  triangle  FlG-  18- 

shall  coincide  exactly  with  AB;  the  triangle  being  below  the 

line. 

Then,  placing  the  thumb  and  fingers  of  the  left  hand  firmly 
on  the  ruler,  slide  the  triangle,  with  the  other  hand,  along  the 
ruler,  until  the  side  which  coincided  with  AB  reaches  the  point 
C.  Extend  the  first  and  second  fingers  of  the  left  hand  upon 
the  triangle  to  hold  it  firmly  in  place,  the  thumb  and  remaining 
fingers  steadying  the  ruler,  and  with  the  right  hand,  mark  with 
a pen  or  pencil,  a line  through  C : this  line  will  be  parallel 
to  AB. 

II.  To  draw  through  a given  point  a line  which  shall  be  per- 

pendicular to  a given  line. 

49.  Let  AB  be  the  given  line,  and  D the  given  point. 

Place  the  hypothenuse  of  the  triangle 

against  the  edge  of  the  ruler,  as  before. 

Then  place  the  ruler  and  triangle  so  that 

A.  D B 

one  of  the  sides  of  the  triangle  shall  coin- 

Fig.  19. 

cide  exactly  with  the  line  AB.  Then  slide 
the  triangle  along  the  ruler  until  the  other  side  reaches  the 
point  D : then,  draw  through  Z),  a right  line,  and  it  will  be 
perpendicular  to  AB. 

The  right  angle  of  the  triangle  should  be  carefully  tested 
by  laying  off  a perpendicular  through  the  same  given  point, 


36 


ELEMENTS  OF  SURVEYING. 


[BOOK  L 


with  the  triangle  in  two  positions— to  the  right  of  the  per- 
pendicular, and  then  to  the  left  of  it. 

50.  A Scale  of  Equal  Parts  (Fig.  20)  is  formed  by  divid- 
ing a line  of  a given  length,  into  equal  portions. 

* 1 t 6 

Fig.  20. 


If,  for  example,  the  line  aby  of  a given  length,  say  one  inch, 
be  divided  into  any  number  of  equal  parts,  as  10,  the  scale  thus 
formed  is  called  a scale  of  ten  parts  to  the  inch.  The  line  ab, 
which  is  divided,  is  called  the  unit  of  the  scale.  This  unit  is 
laid  off  several  times  on  the  left  of  the  divided  line,  and  the 
points  marked  1,  2,  3,  &c. 

The  unit  of  scales  of  equal  parts  is,  in  general,  either  an 
inch,  or  an  exact  part  of  an  inch.  If,  for  example,  ab , the  unit 
of  the  scale,  were  half  an  inch,  the  scale  would  be  one  of  10  parts 
to  half  an  inch,  or  of  20  parts  to  the  inch. 

If  it  were  required  to  take  from  the  scale  a line  equal  to 
two  inches  and  six-tenths,  place  one  foot  of  the  dividers  at  2, 
on  the  left,  and  close  the  other  to  .6,  which  marks  the  sixth 
of  the  small  divisions : the  dividers  will  then  embrace  the 
required  distance. 


Fig.  21. 


51.  A Diagonal  Scale  of  Equal  Parts  (Fig.  21)  is  thus 
constructed  Take  ab  for  the  unit  of  the  scale,  which  may  be 


SEC.  III.] 


INSTRUMENTS  FOR  PLOTTING. 


37 


one  inch,  \,  \,  or  f of  an  inch,  in  length.  On  ab  describe 
the  square  abed.  Divide  the  sides  ab  and  dc  each  into  ten  equal 
parts.  Draw  af \ and  the  other  nine  parallels  as  in  the  figure. 

Produce  ba , to  the  left,  and  lay  off  the  unit  of  the  scale 
any  convenient  number  of  times,  and  mark  the  points  1,  2,  3, 
&c.  Then,  divide  the  line  ad  into  ten  equal  parts,  and  through 
the  points  of  division  draw  parallels  to  ab,  as  in  the  figure. 

Now,  the  small  divisions  of  the  line  ab  are  each  one-tenth  (.1) 
of  ab  ; they  are  therefore  . 1 of  ad,  or  . 1 of  ag  or  gh. 

If  we  consider  the  triangle  adf,  we  see,  that  the  base  df  is 
one-tenth  of  ad,  the  unit  of  the  scale.  Since  the  distance  from 
a to  the  first  horizontal  line  above  ab  is  one-tenth  of  the  distance 
ad,  it  follows  that  the  distance  measured  on  that  line,  between 
ad  and  af,  is  one-tenth  of  df : but  since  one-tenth  of  a tenth  is 
a hundredth,  it  follows  that  this  distance  is  one  hundredth  (.01) 
of  the  unit  of  the  scale.  A like  distance,  measured  on  the  second 
line,  is  two  hundredths  (.02)  of  the  unit  of  the  scale ; on  the 
third,  .03;  on  the  fourth,  .04,  &c. 

If  it  were  required  to  take,  in  the  dividers,  the  unit  of  the 
scale,  and  any  number  of  tenths,  place  one  foot  of  the  dividers 
at  1,  and  close  the  other  to  that  figure  between  a and  b which 
designates  the  tenths.  If  two  or  more  units  are  required, 
the  dividers  must  be  placed  on  a point  of  division  further  to 
the  left. 

When  units,  tenths,  and  hundredths  are  required,  place  one 
foot  of  the  dividers  where  the  vertical  line  through  the  point 
which  designates  the  units,  intersects  the  line  which  designates 
the  hundredths  : then,  close  the  dividers  to  that  line  between  ad 
and  be  which  designates  the  tenths:  the  distance  so  embraced 
will  be  the  one  required. 

For  example,  to  hike  off  the  distance  2.34,  we  place  one  foot 
of  the  dividers  at  l,  and  close  the  other  to  e : and  to  take  off  the 


38  ELEMENTS  OF  SURVEYING.  [BOOK  I. 

distance  2.58,  we  place  one  foot  of  the  dividers  at  p and  close  the 
other  to  q. 

Note  1. — If  a line  is  so  long  that  the  whole  of  it  cannot  be 
taken  from  the  scale,  it  must  be  divided,  and  the  parts  of  it 
taken  from  the  scale  in  succession. 

Note  2. — If  a line  be  given  upon  the  paper,  its  length 
can  be  found  by  taking  it  in  the  dividers  and  applying  it  to 
the  scale. 


c 


52.  A Semicircular  Protractor  (Fig.  22)  is  used  to  lay 
down,  or  protract  angles.  It  may  also  be  used  to  measure 
angles  included  between  lines,  already  drawn  upon  paper. 

It  consists  of  a brass  semicircle,  ABC \ divided  to  half 
degrees.  The  degrees  are  numbered  from  0 to  180,  both  ways  ; 
that  is,  from  A to  B and  from  B to  A.  The  divisions,  in  the 
figure,  are  made  only  to  degrees.  There  is  a small  notch  at 
the  middle  of  the  diameter  AB,  which  indicates  the  centre  of 
the  protractor. 


SEC.  III.] 


INSTRUMENTS  FOR  PLOTTING. 


39 


To  lay  off  an  angle  with  a Protractor. 

53.  Place  the  diameter  AB  on  the  line,  so  that  the  centre 
shall  fall  on  the  angular  point.  Then  count  the  degrees  con- 
tained in  the  given  angle,  from  A toward  B , or  from  B toward 
A,  and  mark  the  extremity  of  the  arc  with  a pin.  Remove 
the  protractor,  and  draw  a line  through  the  point  so  marked, 
and  the  angular  point : this  line  will  make  with  the  given  line 
the  required  angle. 

The  ordinary  brass  or  horn  protractors  are  of  but  little  value. 
Printed  protractors  of  six  and  twelve  inches  diameter,  upon 
heavy  paper,  or  bristol  board,  and  divided  to  quarter  degrees, 
are  very  useful  and  reliable. 

By  the  following  method  an  angle  may  be  laid  off  with  even 
greater  accuracy  than  with  a protractor.  With  a reliable  scale  of 
inches,  divided  to  hundredths,  set  the  dividers  at  five  inches,  and 
placing  one  point  at  the  given  vertex  A (Fig. 

23),  describe  the  arc  BG;  take  out  from  the 
table  the  natural  sine  of  half  the  given  angle, 
multiply  it  by  10,  and  call  the  product  inches  ; 
with  the  dividers  take  off  this  number  of  inches 
from  the  scale,  and  placing  one  point  at  B,  describe  an  arc 
cutting  BC  at  G ; GAB  will  be  the  required  angle. 

It  is  evident  that  twice  the  sine  of  half  the  angle  is  the  chord 
of  the  whole  angle ; and  since  the  radius  used  is  5 inches,  we 
have 


Fig.  23. 


2 (Nat.  sine  \a  x 5)  = 10  x Nat.  sine  \a. 


40 


ELEMENTS  OF  SURVEYING. 


[ROOK  I. 


SECTION  I V. 

DEFINITIONS. 

54.  Surveying  comprises  all  the  operations  necessary  for 
finding  the  lengths  and  directions  of  the  bounding  lines  of  any 
portion  of  the  earth’s  surface,  the  area  of  such  portion,  and  for 
making  on  paper  an  accurate  delineation,  or  map,  of  the  surface 
and  the  boundaries. 

55.  Plane  Surveying  is  that  branch  of  surveying  in  which 
the  curvature  of  the  earth  is  neglected,  as  it  may  be  when  the 
survey  is  limited  to  small  portions  of  the  surface. 

56.  Geodesy,  or  Geodetic  Surveying,  is  that  branch  in 
which  the  curvature  of  the  earth  is  taken  into  account,  as  it 
must  be  in  all  extensive  surveys. 

57.  A Horizontal  Plane  at  any  point  is  a plane  perpen- 
dicular to  the  radius  of  the  earth  drawn  to  that  point. 

58.  A Vertical  Plane  is  a plane  perpendicular  to  a hori- 
zontal plane. 

59.  A Horizontal  Line  is  any  line  of  a horizontal  plane. 

60.  A Vertical  Line  is  a line  perpendicular  to  a hori- 
zontal plane. 

61.  An  Oblique  Line  is  a 
line  inclined;  i.  e.,  neither  par- 
allel nor  perpendicular  to  a hori- 
zontal plane. 

Thus  (Fig.  24),  A B and  DC 
arc  horizontal  lines;  BC  and 

AD  are  vertical  lines  ; and  A C and  BD'QXQ  oblique  lines. 


SEC.  IY.J 


DEFINITIONS. 


41 


62.  The  Horizontal  Distance  between  two  points  is  the 
horizontal  line  intercepted  between  the  two  vertical  lines  passing 
through  those  points.  Thus,  DC  or  AB  (Fig.  24),  is  the  hori- 
zontal distance  between  the  two  points  A and  C,  or  between 
the  points  B and  D. 

63.  A Horizontal  Angle  is  an  angle  whose  sides  are  hori- 
zontal ; the  plane  of  its  sides  is  also  horizontal. 

64.  A Vertical  Angle  is  an  angle  the  plane  of  whose 
sides  is  vertical. 

65.  An  Angle  of  Elevation  is  a vertical  angle  having  one 
of  its  sides  horizontal  and  the  other  oblique,  the  oblique  side 
being  above  the  horizontal  side.  Thus,  BAC  (Fig.  24)  is  the 
angle  of  elevation  from  A to  C . 

66.  An  Angle  of  Depression  is  a vertical  angle  having 
one  of  its  sides  horizontal  and  the  other  oblique,  the  oblique  side 
being  beloiv  the  horizontal  side.  Thus,  DC  A (Fig.  24)  is  the 
angle  of  depression  from  C to  A. 

67.  An  Oblique  Angle  is  an  angle  the  plane  of  whose 
sides  is  inclined  to  a horizontal  plane. 


BOOK  II. 

CHAIN  SURVEYING. 


SECTION  I. 

MEASUREMENT  OF  Dl  STANCES. 

68.  Any  tape,  rod,  or  chain,  divided  into  equal  parts,  may  be 
used  as  a measure  for  finding  the  distance  between  two  points. 


The  measure  in  general  use  for  land  surveying  is  a chain  of 
four  rods,  or  sixty-six  feet  in  length,  called  Gunter’s  chain 
(Pig.  25),  from  the  name  of  the  inventor.  It  is  composed  of  100 
links,  each  joined  to  the  other  by  two  or  three  rings.  Every  tenth 
link  from  either  end,  is  marked  by  a small  attached  brass  pendant 


SEC.  I.] 


MEASUREMENT  OF  DISTANCES. 


43 


or  tag,  which  is  notched  to  designate  its  number  from  the  end. 
The  tag  at  the  middle,  or  fifty-link,  point  is  distinguished  by 
being  rounded,  or  by  some  other  peculiarity  of  make.  As  the 
tags  at  equal  distances  from  the  two  ends  of  the  chain  are 
marked  the  same,  care  must  be  taken  not  to  mistake  forty  links 
for  sixty,  &c.,  and  the  reverse.  To  avoid  such  error,  it  would  be 
better  to  have  the  tags  marked  in  regular  order  from  the  begin- 
ning ' 3 end  of  the  chain,  rather  than  from  both  ends  to  the 
middl 

A link  in,  measure  includes  a bar , with  its  connecting  ring  at 
each  end;  when  there  are  three  connecting  rings,  a ring  and  a 
half  at  each  end  is  included. 

The  handles  &re  of  brass,  and  each  forms  part  of  the  end  link, 
to  which  it  is  connected  by  a nut,  by  which  also  the  length  of 
the  chain  is  adjusted. 

To  determine  whether  to  measure  from  the  inside  of  the  brass 
handle,  or  from  the  outside , double  back  the  last  two  or  three 
links  upon  the  preceding  links  and  compare. 

The  division  of  the  chain  into  100  equal  parts  is  very  con- 
venient, since  the  divisions,  or  links,  are  decimals  of  the  whole 
chain,  and  in  the  calculations  are  treated  as  such. 

TABLE. 

1 chain  = 4 rods  = G6  feet  = 792  inches  = 100  links. 

1 link  = 7.92  inches. 

80  chains  = 320  rods  = 5280  feet  = 1 mile. 

An  excellent  chain  for  accurate  measurements  is  Griim- 
man’s  patent  “suspended  chain,”  which  is  made  of  very  light 
steel  wire,  is  fitted  with  spring-balance,  thermometer  and  spirit- 
level  attachments,  and  is  held  above  the  surface  when  in  use,  the 
ends  of  the  chain  being  marked  upon  the  ground  by  the  points 
of  plummets  let  fall  from  the  end  notches. 


44 


ELEMENTS  OF  SUKVEYING. 


[BOOK  IJ. 


Instead  of  a chain,  which  is  liable  to  error  because  of  the 
bars  and  rings  becoming  worn  by  frequent  contacts,  a steel 
ribbon  or  tape  is  often  used. 

69.  Besides  the  chain  or  tape,  the  surveyor  needs  ten  (or 
better,  eleven)  marking  pins  (Fig.  26),  made  of  iron  or  steel  wire, 
about  an  eighth  of  an  inch  in  thickness  and  a foot  long,  sharpened 
at  one  end  and  bent  into  a ring  at 
the  other,  for  marking  chain 
lengths  on  the  ground  ; a plumbob 
(Fig.  27)  and  line  for  referring, 
when  necessary,  points  in  the  chain 
held  horizontally  to  the  inclined 
surface  of  the  ground  ; and  a set 
of  flag-poles,  or  ranging  rods,  for 
marking  stations  and  ranging  out 
lines.  The  marking  pins  should 
be  strung  upon  an  iron  ring  with 
a spring-catch,  and  this  ring 
should  be  attached  to  a strap  to 
be  passed  over  the  right  shoulder 
suspending  the  pins  at  the  left  side  ; or,  better,  the  pins  may  be 
carried  in  a leather  quiver  strapped  to  the  waist. 

The  pins  should  be  tagged  with  white  cloth  to  enable  the 
surveyor  to  find  them  again  readily,  when  they  have  been  left  to 
mark  a point. 

70.  To  Measure  a Horizontal  Line. — The  point  where 
the  measurement  is  to  begin  is  located  by  a staff  temporarily 
placed  for  the  purpose,  or  by  some  one  of  the  many  permanent 
marks  by  which  the  angular  points  in  a houndary  are  fixed. 

The  other  extremity  of  the  line  must  be  provided  with  a staff 
or  Hag  which  can  be  easily  seen. 

Two  chainincn  are  required,  a fore-chainman,  or  leader,  and 


Fig.  26.  Fig.  27. 


SEC  I.] 


MEASUREMENT  OF  DISTANCES. 


45 


a hind-chainman,  or  follower.  The  more  careful  and  expert  of 
the  two  should  be  the  follower. 

The  leader,  with  the  marking-pins  and  one  handle  of  the 
chain  in  his  right  hand,  starts  off  on  the  line,  drawing  out  the 
chain  to  its  full  length.  Both  chainmen  now  examine  it  to  see 
that  there  are  no  inaccuracies  in  it,  either  from  bent  links  or 
kinks  in  the  rings  joining  the  links.  Having  adjusted  the  chain 
for  use,  the  leader  resumes  his  place,  to  be  directed  by  the 
follower,  who  stands  behind  the  staff  at  the  beginning,  and  sights 
to  the  staff  at  the  end  of  the  line,  so  that  the  measurement  shall 
be  made  exactly  along  the  established  line. 

To  facilitate  this  (on  level  ground)  and  to  insure  the  correct 
alignment  of  the  pin,  at  its  proper  distance,  the  chain  and  one 
pin  should  be  held  firmly  in  the 
right  hand,  as  represented  in  Figure 
28.  While  the  pin  is  being  aligned, 
it  should  he  held  by  the  leader  as  far 
from  the  body  as  possible,  so  that  the 
view  of  the  flag  be  left  unobstructed. 

To  accomplish  this,  and  at  the  same 
time  draw  the  chain  to  the  proper 
degree  of  tension,  the  right  arm 
should  be  braced  against  the  inside 
of  the  right  knee. 

The  follower  directs,  by  the  simple  orders  “ right”  or  “left,” 
according  as  the  pin,  held  as  described,  is  to  be  carried  to  the  right 
or  left  to  bring  it  into  line  with  the  flag.  When  the  pin  is  truly 
in  line,  the  chain  at  the  same  time  being  drawn  straight  and 
taut,  the  order  “ down  ” is  given,  when  the  leader  bringing  his 
left  hand  to  hear  on  the  top  of  the  pin,  forces  it  vertically  into 
the  ground,  and  resumes  his  course  to  the  length  of  another 
chain. 


46 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


If,  for  any  reason,  the  pin  can  not  be 
driven  into  the  ground,  the  end  of  the 
chain  length  should  be  marked  by  driving 
the  pin  obliquely , always  at  right  angles  to 
the  chain ; if  this  cannot  be  done,  a cross 
should  be  scratched  on  the  ground  at  the 
exact  point,  and  the  pin  laid  down  with  its 
point  at  the  mark. 

After  one  or  two'  chains  have  been 
measured,  on  any  line,  the  leader  can,  by 
glancing  back  to  the  station  just  left,  place  the  pin  nearly  in  the 
right  position  ; the  exact  aligning  should  be  left,  however,  to 
the  follower. 

When  the  distance  to  be  measured  is  more  than  ten  chains, 
the  pins,  when  exhausted,  should  be  returned  to  the  leader,  the 
distance  noted  in  a field-book  provided  for  the  purpose,  and  the 
chaining  recommenced  at  the  place  of  the  tenth  pin.  If  but  ten 
pins  are  used,  the  follower  has  then  but  the  hole  made  by  the 
tenth  pin  to  measure  from.  For  this  reason,  eleven  pins  are 
often  used,  the  eleventh  being  of  different  material  from  the 
others  for  distinction,  and  is  used  by  the  follower  to  measure 
from  when  the  ten  pins  are  returned  to  the  leader. 

71.  All  distances  should  be  measured  horizontally.  Hence, 
when  the  ground  slopes,  one  end  of  the  chain  must  be  elevated. 
Each  chainman  should  be  provided  with  a small  plumb-line,  so 
that  the  elevated  end  of  the  chain  may  be  held  directly  over  the 
proper  point. 

When  the  raised  end  of  the  chain  is  only  two,  or  even  three 
feet  above  the  ground,  it  will  suffice,  in  many  cases,  to  use  a 
marking-pin,  held  lightly  by  the  point,  between  the  thumb  and 
finger,  instead  of  a plumb-line.  When  the  chaining  is  on  a steep 
inclination,  other  precautions  should  be  observed. 


Fig.  29. 


SEC.  I.] 


MEASUREMENT  OF  DISTANCES. 


4? 


Suppose  the  chaining  to  be  up  hill.  The  leader  draws  the 
chain  out  to  its  full  length , as  in  any  other  case,  and  then  returns 


to  within  such  a distance  of  the  follower,  that  when  the  chain  is 
drawn  out  to  that  length  horizontally,  it  shall  not  be  too  high  to 
be  held  conveniently. 

The  follower  holds  his  end  of  the  chain  carefully  over  the 
point  or  station,  by  means  of  the  plumb-line,  while  he  directs  the 
leader  in  the  usual  manner. 

The  point  fixed  in  this  manner,  by  the  leader,  must  not  be 
marked  by  a marking-pin,  but  by  a small  peg  or  nail.  At  the 
order,  “Down,”  the  leader  does  not  go  forward  immediately,  but 
waits  until  the  follower  comes  up  and  takes  the  chain  by  the 
precise  point  held,  the  moment  before,  to  the  ground. 

This  point  is  now  held  above  the  peg  by  the  the  follower, 
who  uses  the  plumb,  as  before,  and  aligns  the  leader,  who  has 
taken  hold  of  the  chain  a few  links  farther  on,  and  is  holding  it 
to  the  ground.  These  short  distances  are  not  recorded.  The  end 
of  a full  chain  is  marked  by  a marking-pin. 


48 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


Iii  chaining  doion-liill , the  method  is  essentially  the  same*. 
The  leader  uses  the  plumb,  and  determines  by  it  where  the  peg 
is  to  be  placed. 

At  the  end  of  a course,  the  part  of  a chain  is  measured  by 
drawing  the  chain  only  to  the  flag , where  it  is  held  by  the  leader, 
until  the  follower  comes  forward  to  the  last  pin,  and  counts 
the  links. 

In  ifieasuring  up  the  hill  from 
A to  C,  or  down  the  hill  from 
C to  A,  the  horizontal  distances 
a b,  c d,  and  / C\  are  measured  and 
their  sum  is  the  horizontal  dis- 
tance between  A and  0. 

Chaining  doivn  hill  gives  more 
accurate  results  than  chaining  up  ; 
follower  holds  the  chain  firmly  upon  the  ground  and  no  ordinary 
pull  by  the  leader  moves  it.  It  is  impossible  to  hold  a chain 
perfectly  steady  over  a point,  by  means  of  a long  plumb-line, 
while  the  leader  is  pulling  out. 

72.  When  the  ends  of  a line  can  not  be  seen,  each  from  the 
other,  intermediate  points  between  the  two  must  be  established. 
When  a hill  intervenes,  such  points  may  be  established  thus  : let 
the  surveyor  and  an  assistant,  each  with  a ranging-rod,  place 
themselves  as  nearly  in  the  line  as  possible,  and  in  such  position 
that  each  can  see  the  other  and  the  flag  beyond  him.  The  sur- 
veyor looking  to  the  flag  at  the  end  of  the  line,  directs  the 
assistant  into  line  with  it ; the  assistant  then  looks  to  the  flag 
at  the  beginning  of  the  line  and  directs  the  surveyor  into 
line  with  it ; the  surveyor  from  his  new  position  redirects 
the  assistant  into  line  with  the  end  flag-staff ; the  assistant 
then  realigns  the  surveyor  with  the  flag-staff  at  the  beginning 
of  the  line  ; the  operation  is  repeated  till  both  stand  in  the 


SEC.  I.J 


MEASUREMENT  OF  DISTANCES. 


49 


desired  line,  when  their  positions  are  marked  with  the  rang- 
ing-rods. 

73.  When  a valley  is  to  be  chained  across,  intermediate 
points  in  the  lower  portions  of  it  may  be  fixed,  if  necessary,  by 
the  surveyor  holding  a plumb-line  so  as  to  cover  the  flag-staffs  at 
both  ends  of  the  line  and  directing  an  assistant  to  fix,  between 
the  two,  ranging-rods  which  shall  also  be  covered  by  the  plumb- 
line. 

74.  When  a wood  intervenes  between  the  two  ends  of  a line, 
a trial  line  may  be  run  out  by  ranging-rods  placed  at  convenient 
distances  in  line  with  each  other  and  with  the  staff  at  the 
beginning  of  the  line,  and  as  nearly  as  possible  in  the  required 
line.  Then  draw  on  the  ground  a perpendicular  (by  a method 
to  be  shown  presently)  from  the  staff  at  the  end  of  the  required 
line  to  the  trial  line  just  run  out,  and  measure  the  length  of 
this  perpendicular.  The  ranging-rods  may  then,  by  the  property 
of  similar  triangles,  be  put  in  their  true  position  in  line.  Thus, 


A G is  to  be  measured ; the  trial  line  AF  is  run  out  and  the 
ranging-rods  B,  C,  &c.,  fixed  at  known  distances  apart  and  as 
nearly  in  the  line  as  possible  ; the  perpendicular  GF  is  measured  ; 
then  from  the  similar  triangles,  AFG  and  ABH , 

AF:  AB  ::  FG  : BH. 

The  distance  BII  thus  becomes  known,  and  the  ranging-rod 
at  B is  moved,  on  a perpendicular  to  AF,  the  required  distance 
to  its  true  position  at  II.  The  other  ranging-rods  are  in  like 


50 


ELEMENTS  OF  SURVEYING. 


[ROOK  II. 


manner  put  in  their  true  positions  at  1 , K , &c.,  and  the  true  lino 
is  marked  out. 

75.  To  trace  on  the  ground  the  direction  of  a straight  line, 
that  shall  be  perpendicular,  at  a given  point,  to  a given 
straight  line. 

FIRST  METHOD. 

Any  three  lines  having  the  ratio  3,  4,  and  5,  form  a right- 
angled  triangle. 

Let  AB  (Fig.  33)  be 
the  given  line  and  C the 
point  at  which  the  per- 
pendicular is  to  be  drawn. 

Divide  the  number  of 
links  in  the  chain  by  8, 
neglecting  the  remainder ; 
if  we  are  using  the  100-link  chain  the  quotient  would  be  12  links. 
From  the  point  C measure  a distance  towards  A equal  to  four 
times  this  quotient  (48  links);  place  one  end  of  the  chain  at  C, 
and  the  end  of  the  96th  link  [(5  + 3)  x 12  = 96]  at  + ; then 
taking  the  end  of  the  36th  link  (3  x 12)  pull  out  the  chain  so 
that  the  two  portions  EA  and  EC  are  taut,  and  E will  be  a 
point  on  the  perpendicular  required. 

This  method  supposes  the  chain  to  be  correctly  divided  into 
links. 

SECOND  METHOD. 

Let  AD  be  the  given  right  line, 
and  D the  point  at  which  the  perpen- 
dicular is  to  be  drawn.  Take  the 
longest  available  distance  on  the  tape 
or  chain  (the  whole  of  it  if  possible), 
and  place  one  extremity  at  D,  and 
fasten  the  other  at  some  point,  as  E, 


E 


Fig.  34. 


SEC.  I.] 


MEASUREMENT  OF  DISTANCES. 


51 


between  the  two  lines  which  are  to  form  the  right  angle.  Place 
a staff  at  E.  Then,  having  stationed  a person  at  D,  remove  that 
extremity  of  the  chain  and  carry  it  round  until  it  ranges  on 
the  line  DA,  at  A.  Place  a staff  at  A : then  remove  the  end 
- of  the  chain  at  A,  and  carry  it  round  until  it  falls  on  the  line 
AE,  prolonged,  at  F.  Then  place  a staff  at  F;  ADF  will  be 
a right  angle,  being  an  angle  in  a semicircle. 

This  method  is  independent  of  any  errors  of  graduation  of 
the  chain  ; it  also  gives  the  largest  possible  construction  in  the 
field,  a matter  of  importance  as  insuring  the  most  correct 
results. 


76.  There  is  a simple  instrument  for  laying  off  right  angles  on 
the  ground  called  the  Surveyor’s 
Cross.  This  instrument  con- 
sists of  two  bars,  AB  and  CD, 

Fig.  35,  permanently  fixed  at 
right  angles  to  each  other,  and 
firmly  attached  at  E,  to  a 
pointed  staff,  which  serves  as 
a support.  Four  sights  are 
screwed  firmly  to  the  bars,  by 
means  of  the  screws  a,  b,  c, 
and  d. 

As  the  only  use  of  this  in- 
strument is  to  lay  off  right 
angles,  it  is  of  the  first  im- 
portance that  the  lines  of  sight 

be  truly  at  right  angles.  To  ascertain  if  they  are  so,  let  the  bar 
AB  be  turned  until  its  sights  mark  some  distinct  object ; then 
look  through  the  other  sights,  and  place  a staff  on  the  line  which 
they  indicate ; let  the  cross  be  then  turned  until  the  sights 
of  the  bar  AB  come  to  this  last  line;  if  the  other  sights  are 


52 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


directed  to  the  first  object,  the  lines  of  sight  are  exactly  at 
right  angles. 

The  sights  being  at  right  angles,  if  one  of  them  be  turned 
in  the  direction  of  a given  line,  the  other  will  mark  the  direc- 
tion of  a line  perpendicular  to  it,  at  the  point  where  the  instru- 
ment is  placed. 

77.  From  a given  point  without  a straight  line,  to  let  fall  a 
perpendicular  on  the  line. — Let  C be  the  given  point,  and  AB 
the  given  line  (Fig.  30). 

From  G,  measure  a line,  as  CA, 
to  any  point  of  the  line  AB.  From 
A,  measure  on  AB  any  distance  as 
AF \ and  at  F erect  FE  perpendicu- 

1 1 Fig.  36. 

lar  to  AB. 

Having  stationed  a person  at  A,  measure  along  the  perpen- 
dicular FE  until  the  forward  staff  is  aligned  on  the  line  AC: 
then  measure  the  distance  AE.  From  similar  triangles, 

AE  : AF  ::  AC  : AD ; 

in  which  all  the  terms  are  known  except  AD,  which  may,  there- 
fore, be  found.  The  distance  AD  being  laid  off  from  A,  the 
point  D,  at  which  the  perpendicular  CD  meets  AB,  becomes 
known.  If  the  length  of  the  perpendicular  is  desired,  it  may  be 
found  from  the  proportion, 

AE:  EF  ::  AC:  CD, 

in  which  all  the  terms  are  known  except  CD. 

It  is  always  best  to  make  AF  nearly  equal  to  AD  when 
practicable;  for  then  AE  will  be  nearly  equal  to  AC,  and  the 
multiplication  of  errors  will  be  avoided ; in  the  expression 

AD  = 'J  AF,  A ('  will  be  but  little  greater  than  1. 


SEC.  I.] 


MEASUREMENT  OF  DISTANCES. 


53 


78.  To  trace  on  the  ground  a straight  line  that  shall  pass 
through  a given  point  and  he  parallel  to  a given  straight  line. 

Let  AB  be  the  given 
line  and  P the  given 
point.  From  P meas- 
ure any  oblique  line  to 
AB,  as  PQ,  and  mark 
its  middle  point,  which 
call  m in  the  figure.  From  any  point  of  AB,  as  R,  run  a line 
through  m,  and  prolong  it  till  mS  = Rm ; then  PS  will  be  the 
parallel  required. 


79.  To  determine  the  horizontal  distance  from  a given  point 
to  an  inaccessible  object. 

Let  A be  an  inaccessible  object, 
and  E the  point  from  which  the  dis- 
tance is  to  be  measured. 

First  Method. — At  the  point  E , 
lay  off  EB  perpendicular  to  the  line 
EA,  and  measure  along  it  any  con- 
venient distance,  as  EB. 

At  B lay  off  the  right  angle  EBD,  and  measure  any  distance 
in  the  direction  BD.  Let  a person  at  D align  a staff  on  DA, 
while  a second  person  at  B aligns  it  on  BE : the  staff  will  thus 
be  fixed  at  C.  Then  measure  the  distance  BC. 

The  two  triangles  BCD  and  CA E being  similar,  we  have, 


D 


Fig 


BC  : BD  ::  CE  : EA, 

in  which  all  the  terms  are  known,  except  the  fourth,  which  is, 
therefore,  found. 

If  BC  can  be  made  equal  to  CE,  then  the  measured  distance 
BD  is  the  distance  required.  The  triangle  BDC  should  be  as 


54  ELEMENTS  OF  SURVEYING.  [liOOK  II. 

nearly  equal  to  ACE  as  practicable,  and  the  angle  BCD  should 
not  exceed  45°. 

Second  Method.— Let  B be  the  given  point,  and  A the 
inaccessible  object ; it  is  required  to  find  BA. 

Measure  any  horizontal  base-line, 
as  BC.  Then,  having  placed  staves 
at  B and  C,  measure  any  conve- 
nient distances  BD  and  CE,  such 
that  the  points  D,  B , and  A , shall 
be  in  the  same  right  line,  as  also, 
the  points  E,  C,  and  A ; then  meas- 
ure the  diagonal  lines  DC  and  EB. 

Now,  in  the  triangle  BEC,  the 
three  sides  are  known,  therefore,  the  angle  ECB  can  be  found. 
In  the  triangle  CDB , the  three  sides  are  also  known,  therefore, 
the  angle  CBD  can  be  determined.  These  angles  being  respec- 
tively subtracted  from  180°,  the  two  angles  ACB  and  ABC 
become  known;  and  hence,  in  the  triangle  ABC , we  have  two 
angles  and  the  included  side,  to  find  the  side  BA. 

The  lines  BD  and  CE  should  each  equal  BC,  if  possible,  to 
facilitate  computation,  and  the  angle  at  A should  not  be  less 
than  40°. 

Third  Method. — Let  AC  be  the 
distance  required.  Lay  off  the  right 
angle  CAB , and  measure  AB,  any 
convenient  distance.  At  B lay  off  the 
right  angle  CBD,  and  fix  the  point  D, 
carefully,  in  line  with  AC.  Measure 
AD.  Then, 

AB 2 

AD  : AB  ::  AB  : AC,.'.AC=-.n 

A U 

(Legendre,  Bk.  IV,  Prop.  23). 

Note. — When  such  problems  occur 


Fio.  40. 


SEC.  I.] 


MEASUREMENT  OF  DISTANCES. 


55 


in  practice,  the  distance  A C is  usually  a portion  of  a longer  line, 
so  that  the  line  GAD  is  well  marked  by  stakes  or  pins,  before  AB 
is  measured. 

80.  To  prolong  a line  beyond  an  obstacle. — If  the  obstacle 
can  be  seen  over,  the  surveyor  should  send  an  assistant, 
with  a flag-pole  or  ranging-rod,  to  the  further  side  of  it,  to 
a pbint  approximately  in  line.  At  this  point  the  assistant 
should  hold  the  rod  vertical  and  exposed  to  the  surveyor,  at  one 
end  of  the  line,  who  directs  him  to  “right”  or  “left”  till  the 
ranging-rod  covers  or  coincides  with  the  flag-pole  at  the  further 
end  of  the  line,  when  it  is  inserted  in  the  ground  and  marks  a 
point  in  the  desired  line.  Other  points  may  be  determined  in 
like  manner.  This  is  called  “ranging,”  or  “ranging  out”  the 
line. 

When  the  obstacle  cannot  be  seen  over,  the  continuation  may 
be  effected  as  follows  : 

First  Method. — Let  OA 
be  the  line  to  be  prolonged. 

Lay  off  OAB  = 120°,  or 
CAB  = 60°.  Measure  AB , 
of  such  length  as  to  permit 
BC  to  be  measured  without 
meeting  the  obstruction. 

Make  ABC  — 60°,  and 
measure  BC,  equal  to  AB. 

If  A be  not  in  sight  from  C, 
make  the  angle  BCP  equal  to  120°,  and  resume  the  survey  of 
the  line.  A C is  equal  to  AB  or  BC. 

Note. — This  method  may  be  employed  in  the  absence  of  any 
angular  instruments,  by  constructing  an  equilateral  triangle  with 
the  chain.  Measure  half  a chain  from  A towards  C;  then  fasten 


56 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


the  ends  of  the  chain  at  A and  the  point  so  determined,  and  pull 
out  the  middle,  thus  forming  an  equilateral  triangle.  If  there  is 
not  room  to  measure  towards  C,  measure  back  towards  0,  and 
construct  the  angle  on  the  side  of  A 0 away  from  B.  The  errors 
are  cumulative  in  this  method,  but  it  is  rapid,  and  will  do  for 
work  not  requiring  great  accuracy. 

Second  Method. 

Take  two  points,  A 
and  B,  at  convenient 
distance  apart,  one  or 
two  chains,  and  draw 

Fio.  42. 

two  offsets,  AE  and 

BF,  at  right  angles  to  the  direction  of  the  line  and  of  sufficient 
length  to  clear  the  obstacle  ; draw  EF  through  the  extremities  of 
the  rectangular  offsets  and  prolong  it  beyond  the  obstacle  ; this 
last  line  will  be  parallel  to  the  original  line  of  direction  ; at  G 
and  H draw  the  lines  at  right  angles  to  EH,  and  make  them 
equal  in  length  to  A E and  BF;  draw  CD,  it  will  be  the  pro- 
longation required.  FG  is  equal  in  length  to  BC,  provided  the 
perpendiculars  BF  and  GO  are  accurately  laid  out.  This  is  a 
good  method  to  use  in  passing  trees  or  small  obstacles,  provided 
the  distance  between  perpendiculars  AE  and  BF,  also  CG  and 
DH,  is  one  chain  or  more.  The  perpendiculars  are  often  laid  off 
by  guess,  which  method  will  give  a very  fair  prolongation,  but 
the  distance  BG  thus  obtained  will  not  be  accurate. 


Third  Method. — From  any  point  A,  measure  AB;  through 
its  middle  point#,  run  GD,  making  Dx  = Gx;  then  DB  will  be 


SEC.  I.]  MEASUREMENT  OF  DISTANCES.  57 

parallel  to  AC.  From  D run  a line,  DE,  through  any  point  of 
the  line  AB , as  y,  making  yE  of  such  length  that, 

By  : Dy  ::  Ay  : Ey. 

Then  through  z,  the  middle  point  of  DE,  run  BH,  making 
zH  — Bz.  HE  will  be  the  prolongation  of  A C. 

To  find  CH  we  have, 

By  : BD  : : Ay  : AE, 
from  which  AE  is  known  ; 

AE—(AC+  HE)  = CH. 


81.  To  find  the  altitude  of  an  object,  when  the  distance  to 
the  vertical  line  passing  through  the  top  of  it  is  known. — Let 

CD  be  the  altitude  required,  and  A C the  known  distance. 


From  A,  measure  on 
the  line  A C,  any  conveni- 
ent distance  AB,  and  place 
a staff  vertically  at  B.  - 
Then  placing  the  eye  at  A, 
sight  to  the  object  D,  and 
let  the  point,  at  which  the 
line  AD  cuts  the  staff  BE, 
BE  on  the  staff ; then, 


be  marked.  Measure  the  distance 


AB  : BE  ::  AC:  CD, 


whence  CD  becomes  known. 

If  the  line  A C cannot  be  measured,  on  account  of  intervening 
objects,  it  may  be  determined  by  calculation,  as  in  the  preceding 
article,  and  then,  having  found  the  horizontal  distance,  the 
vertical  line  is  readily  determined,  as  before. 


82.  Standard. — As  the  chain  varies  in  length  from  changes 
of  temperature  and  from  use,  it  should  he  daily  compared  with 


58 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


a standard  kept  for  the  purpose.  A convenient  standard  for 
such  comparison  may  be  made  by  driving  into  a level  and  even 
piece  of  ground  two  stakes,  sawed  off  even  with  the  surface  of 
the  ground,  distant  from  each  other  one  chain,  or  66  feet, 
accurately  measured,  with  nails  driven  into  the  heads  of  the 
stakes  to  mark  the  exact  length  of  the  standard.  Marks  made 
upon  the  coping  of  a wall,  or  a curb-stone,  will  answer  the  same 
purpose. 

If  it  is  found  that  any  line  has  been  measured  with  either  too 
long  or  too  short  a chain,  the  true  distance  may  be  found  by  the 
proportion  ; 

The  length  of  the  standard 
: the  length  of  the  incorrect  chain  used 
: : the  measured  distance 
: the  true  distance. 

For  areas  the  proportion  would  be  ; 

The  square  of  the  length  of  the  standard 
: the  square  of  the  length  of  the  chain  used 
: : the  area  found 
: the  true  area. 


SECTION  II. 

AREA  OR  CONTENTS  OF  GROUND. 

83.  The  surface  of  ground  being,  in  general,  broken  and 
uneven,  it  is  impossible,  without  great  trouble  and  expense, 
to  ascertain  its  exact  area  or  contents.  To  avoid  this  incon- 
venience, it  has  been  agreed  to  refer  every  surface  to  a hori- 
zontal plane;  that  is,  to  regard  all  its  bounding  lines  as  hori- 


SEC.  ll.j 


AREA  OR  CONTENTS  OF  GROUND. 


59 


zontal,  and  its  area  as  measured  by  that  portion  of  the 
horizontal  plane  which  the  boundary  lines  enclose. 

For  example,  if  ABCD  were  a 
piece  of  ground,  having  an  uneven 
surface,  the  whole  would  be  referred 
to  a horizontal  plane,  and  that  part 
of  the  plane  which  is  included  be- 
tween the  bounding  horizontal  lines 
AB,  BC , CD,  DA,  be  taken  for  the 

’ ’ Fig.  45. 

measure  of  the  area. 

In  estimating  land  in  this  manner,  the  sum  of  the  areas  of 
all  the  parts,  into  which  a tract  may  be  divided,  is  equal  to  the 
area,  estimating  it  as  an  entire  piece ; but  this  would  not  be 
the  case  if  the  areas  of  the  parts  had  reference  to  the  actual 
surface,  and  the  area  of  the  whole  were  calculated  from  its 
bounding  lines. 

84.  The  unit  of  measure  of  any  quantity  is  a quantity  of 
the  same  kind,  regarded  as  a standard.  For  lines,  the  unit  is  a 
right  line  of  a known  length,  as  1 foot,  1 link,  1 chain,  or  any 
other  fixed  distance.  In  measuring  land,  the  length  of  Gunter’s 
chain  is  generally  taken  as  the  unit  of  linear  measure. 

85.  The  unit  of  measure  for  surfaces  is  a square  described 
on  the  unit  of  linear  measure. 

When,  therefore,  the  linear  measures  are  feet,  yards,  rods,  or 
chains,  the  superficial  measures,  are  square  feet,  square  yards, 
square  rods,  or  square  chains  ; and  the  numerical  expression  for 
the  area,  is  the  number  of  times  which  the  unit  of  superficial 
measure  is  contained  in  the  land  measured. 

An  Acre,  which  is  the  common  unit  of  measure  for  land,  is 
a surface  equal  in  extent  to  10  square  chains  ; that  is,  equal  to  a 
rectangle  of  which  one  side  is  ten  chains  and  the  other  side  one 
chain. 


60 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


A Rood,  is  one  quarter  of  an  acre. 

Since  the  chain  is  four  rods  in  length,  1 square  chain  con- 
tains  16  square  rods ; and  therefore,  an  acre,  which  is  10  square 
chains,  contains  160  square  rods,  and  a rood  contains  40  square 
rods.  A square  rod  is  called  a perch, 

86.  Land  is  generally  computed  in  acres,  roods,  and  perches, 
which  are  respectively  designated  by  the  letters  A.  R.  P. 

When  the  linear  dimensions  of  a survey  are  chains  or  links, 
the  area  will  be  expressed  in  square  chains  or  square  links,  and  it 
is  necessary  to  form  a rule  for  reducing  such  area  to  acres,  roods, 
and  perches.  The  reduction  may  be  made  by  the  following 

TABLE. 


Miles. 

Acres. 

Roods. 

Sq.  Chains. 

Perches. 

Sq.  Links. 

1 

640 

2560 

6400.0 

102,400 

64,000,000 

1 

4 

100 

160 

100,000 

1 

2.5 

40 

25,000 

1.0 

16 

10,000 

1 

625 

1 square  mile  = 6400  square  chains  = 640  acres. 


When  the  linear  dimensions  are  links,  the  area  will  be  ex- 
pressed in  square  links,  and  may  be  reduced  to  acres  by  dividing 
by  100,000,  the  number  of  square  links  in  an  acre  ; that  is,  by 
pointing  off  five  decimal  places  from  the  right  hand. 

If  the  decimal  part  be  then  multiplied  by  4,  and  five  places  of 
decimals  pointed  off,  in  the  product,  from  the  right  hand,  the 
figures  to  the  left  will  express  the  roods. 

If  the  decimal  part  of  this  result  be  now  multiplied  by  40, 
and  five  places  for  decimals  pointed  off,  as  before,  the  figures  to 
the  left  will  express  the  perches. 

If  one  of  the  dimensions  he  in  links,  and  the  other  in  chains, 


SEC.  II.] 


AREA  OR  CONTENTS  OF  GROUND. 


61 


the  chains  may  be  reduced  to  links  by  annexing  two  ciphers  ; or, 
the  multiplication  may  be  made  without  annexing  the  ciphers, 
and  the  product  reduced  to  acres  and  decimals  of  an  acre,  by 
pointing  off  three  decimal  places  from  the  right  hand. 

When  both  the  dimensions  are  in  chains,  the  product  is 
reduced  to  acres  by  dividing  by  10,  or  pointing  off  one  decimal 
place. 

From  which  it  is  evident  that, 

1st.  If  links  be  multiplied  by  links , the  product  is  reduced  to 
acres  by  pointing  off  five  decimal  places  from  the  right  hand. 

2d.  If  chains  be  multiplied  by  links,  the  product  is  reduced  to 
acres  by  pointing  off  three  decimal  places  from  the  right  hand. 

3d.  If  chains  be  multiplied  by  chains,  the  product  is  reduced 
to  acres  by  pointing  off  one  decimal  place  from  the  right  hand. 

87.  Since  there  are  16.5  feet  in  a rod,  a square  rod  is  equal 
to 

16.5  x 16.5  = 272.25  square  feet. 

If  the  last  number  be  multiplied  by  160,  there  will  result 
272.25  x 160  = 43560  = the  square  feet  in  an  acre. 

Since  there  are  9 square  feet  in  a square  yard,  the  last  number 
divided  by  9,  will  give 

4840  = the  number  of  square  yards  in  an  acre. 

88.  To  find  the  area  of-  a piece  of  ground  in  the  form  of  \ 
square,  rectangle,  or  parallelogram. 

Rule. — Multiply  the  base  by  the  altitude,  and  tht 
product  will  express  the  area  (Geom.,  Bk.  IV.,  Prop.  IV.). 


62 


ELEMENTS  OF  SURVEYING. 


[BOOK  II. 


Example. — To  find  the  area,  of  the  rec-  D| — 
tangular  field  A BCD. 

Measure  the  two  sides  AB,  BC j suppose 
that  AB  =.  14  chains  27  links,  and  BC  = 9 fig.  46. 

chains  75  links.  Then, 

AB  - 1427  links, 

BC  = 975  links, 

ABy.BC  — 1391325  square  links, 

= 13.91325  acres. 

4 

3.65300  roods, 

40 

26.12000  perches. 

Am.  13  A.  3 R.  26  P. 


89.  To  find  the  contents  of  a piece  of  land  in  the  form  of  a 

triangle. 

FIRST  METHOD. 

Rule. — Measure  either  side  of  the 
triangle  as  BC,  and  from  the  opposite 
angle  A,  let  fall  a perpendicular  AD, 
and  measure  this  perpendicular ; 
then,  multiply  the  base  and  perpen- 
dicular together,  and  divide  the 
product  by  2 ; the  result  will  express  the  area  of  the 
triangle  (Geom.,  Bk.  IV.,  Prop.  VI.). 

* EXAMPLES. 

1.  What  are  the  contents  of  a triangle  whose  base  is  25  ch. 
1 1.,  and  perpendicular  18  ch.  14  1.  ? Ans.  22  A.  2 R.  29  P. 


SEC.  II.]  AREA  OR  CONTENTS  OF  GROUND.  63 

2.  What  are  the  contents  of  a triangle  whose  base  is  15.48 
chains,  and  altitude  9.67  chains?  Ans.  7 A.  1 R.  38  P. 

SECOND  METHOD. 

Rule. — Measure  the  three  sides  of  the  triangle.  Then> 
add  them  together  and  take  half  their  sum.  From  this 
half  sum  subtract  each  side  separately . Then,  multiply 
the  half  sum  and  the  three  remainders  together,  and 
extract  the  square  root  of  the  product ; the  result  will  be 
the  area  (Geom.  Mens.,  Art.  97). 

Or,  after  having  obtained  the  three  remainders,  add 
together  the  logarithm  of  the  half  sum  and  the  logarithms 
of  the  respective  remainders,  and  divide  their  sum  by  2 ; 
the  quotient  will  be  the  logarithm  of  the  area. 

EXAMPLES. 

1.  Find  the  area  of  a triangular  piece  of  ground  whose  sides 
are  20,  30,  and  40  chains. 

BY  FIRST  RULE. 

45  45  45 

-20  —30  —40 

25  1st  rem.  15  2d  rem.  5 3d  rem. 

half  sum. 

Then,  45  x 25  x 15  x 5 = 84375 ; 

and  a/84375  = 290.4737  = the  area. 

Ans.  29  A.  0 R.  8 P. 

2.  What  is  the  area  of  a triangle  whose  sides  are  2569,  4900, 
and  5035  links? 


20 
30 
40 
2)  90 
45  = 


64 


ELEMENTS  OF  SURVEYING. 


[rook  II 


BY  SECOND  RULE. 


2569  6252  6252  6252 


4900  —2569  —4900  —5035 

5035  3683  1st  rem.  1352  2d  rem.  1217  3d  rem. 


2 ) 1 250  I 

6252  = half  sum. 


. 3.796019 

. 3.566202 

. 3.130977 

. 3.085291 

2 ) 13.578489 
Area  in  square  links,  6155225  . . . 6.789244 

Ans.  61  A.  2 R.  8 P. 


Then, 


log  6252 
log  3683 
log  1352 
log  1217 


90.  To  find  the  area  of  a piece  of  land  in  the  form  of  a 

trapezoid. 

Rule. — Measure  the  two  parallel  sides,  and  also  the  per- 
pendicular distance  between  them.  Add  the  two  parallel 
sides  together,  and  take  half  the  sum  ; then  multiply  the 
half  sum  by  the  perpendicular,  and  the  product  will  be  the 
area  (Geom.,  Bk.  IV.,  Prop.  VII.). 

EXAMPLES. 

1.  What  is  the  area  of  a trapezoid,  of 
which  the  parallel  sides  are  30  and  49  chains, 
and  the  perpendicular  distance  between  them 
16  eh.  60  1.,  or  16.60  chains  ? 


Fia.  48. 


SEC.  II.] 


AREA  OR  CONTENTS  OF  GROUND. 


65 


30-f-49  — 79;  dividing  by  2,  gives  . . . 39.5 


Multiply  by 16.60 

Area  in  square  chains 655. 700 


Ans.  65  A.  2 R.  11  P. 

2.  Required  the  contents,  when  the  parallel  sides  are  20  and 
32  cli.,  and  the  perpendicular  distance  between  them  26  ch. 

Ans.  67  A.  2 R.  16  P. 

91.  To  find  the  area  of  a piece  of  land  in  the  form  of  a 

quadrilateral. 

Rule. — Measure  the  four  sides  of  the  quadrilateral, 
and  also  one  of  the  diagonals ; the  quadrilateral  will  thus 
be  divided  into  two  triangles,  in  both  of  which  all  the  sides 
will  be  known.  Then,  find  the  areas  of  the  triangles 
separately,  and  their  sum  will  be  the  area  of  the  quad- 
rilateral. _ 

Example. — Suppose  that  the  sides 
and  diagonal  A C,  of  the  quadrilateral 
A BCD  have  been  found, 

AB  =.40.05  ch.,  CD  = 29.87  ch., 

BC  = 26.27  ch.,  AD  = 37.07  ch., 
and  AC  = 55  ch.; 

required  the  area  of  the  quadrilateral. 

Ans. 

Note. — Instead  of  measuring  the  four  sides  of  the  quadri- 
lateral, the  perpendiculars  Bb,  Dg,  may  be  let  fall  on  the 
diagonal  AC.  The  area  of  the  triangle  may  then  be  determined 
by  measuring  these  perpendiculars  and  the  diagonal  AC.  The 
perpendiculars  are  Dg  = 18.95  ch.,  and  Bb  = 17.92  ch. 


I) 


101  A.  1 R.  15  P. 


ELEMENTS  OF  SURVEYING. 


66 


[book  II. 


92.  To  find  the  contents  of  a field  having  any  number  of 

sides. 

Rule. — Measure  the  sides  of  the  field  and  also  the 
diagonals ; the  three  sides  of  each  of  the  triangles  into 
which  the  field  will  be  thus  divided  will  then  be  known , 
and  the  areas  of  the  triangles  may  then  be  calculated  by 
the  preceding  rules.  Or,  measure  the  diagonals,  and  from 
the  angular  points  of  the  field  draw  perpendiculars  to  the 
diagonals  and  measure  their  lengths ; the  base  and  per- 
pendicular of  each  of  the  triangles  will  then  be  known. 


Example. — Let  it  be  required  to  determine  the  contents 
of  the  field  ABCDE having  five  sides. 


Suppose  that  the  diagonals  and  per- 
pendiculars have  been  found, 

AC—  36.21  ch.,  EC  = 39.11  ch., 
Bb=  4.08  ch.,  Dd=  7.26  ch., 

A a = 4.19  ch. ; 


D 


required  the  area  of  the  field. 


Area  of  triangle  ABC  = 73.8684  square  chains, 

Area  of  “ 00^=  141.9693  “ 

Area  of  “ ACE  = 81.7399  “ 

Area  of  “ ABCDE  = 297.5776  “ 

Ans.  29  A.  3 R.  1 P. 


93.  To  determine  the  area,  when  the  diagonals  and  per- 
pendiculars cannot  be  measured ; as  in  the  case  of  swamp  or 
submerged  meadow. 

Rule.  Measure  the  bounding  lines  of  the  area,  and 
then  determine  the  diagonals  by  outside  tie-lines. 


SEC.  II.] 


AREA  OR  CONTENTS  OF  GROUND. 


67 


Thus,  let  ABODE  repre- 
sent the  polygon  including 
the  area,  and  suppose  its 
sides  to  be  measured.  Pro- 
long BA  to  x,  making  Ax 
any  exact  part  of  AB — say 
one  third — and  also  prolong 
EA  to  y , Ay  being  one 
third  of  AE ; then,  because 
of  similar  triangles,  the  measured  length  of  xy  will  be  one  third 
of  BE.  In  like  manner  CE  may  be  determined.  Having  the 
perimeter  and  the  diagonals  proceed  as  in  Art.  92. 

94.  To  find  the  contents  when  the  boundary  is  an  irregular 

line. 

It  frequently  happens  that 
a plot  to  be  surveyed  is 
bounded  partly  by  an  irregu- 
lar line.  In  such  a case, 
one  or  more  straight  lines 
are  surveyed,  and  offsets 
measured  from  these  lines, 
as  often  as  may  be  required 
to  afford  data  for  the  compu- 
tation of  the  area  and  a true 
delineation  of  the  boundary.  In  the  case  represented  in  the 
figure,  the  stream  from  A to  O is  the  boundary.  The  station 
B is  selected  for  convenience,  as  it  is  evident,  if  the  line  were 
run  direct  from  A to  C,  the  labor  of  taking  the  offsets  would  be 
much  greater. 

It  will  be  observed  that  the  offsets  are  so  measured  as  to 
indicate  the  abrupt  bends  in  the  boundary;  and  furthermore,  so 


B 


68  ELEMENTS  OF  SURVEYING.  [HOOK  II. 

that  the  areas  thus  cut  off  may  be  considered  as  being  bounded 
by  straight  lines,  without  sensible  error. 


When  the  boundary  is  a 
crooked  stream  that  is  easily 
crossed,  it  is  often  convenient 
to  survey  a line  across  the 
bend,  as  in  the  figure,  and- 
locate  by  offsets  upon  both 
sides  of  the  line.  In  any 
case,  the  small  areas  to  be 
computed  are  only  trapezoids 
and  triangles. 


95.  To  find  the  area  of  a piece  of  ground  in  the  form  of  s 

circle. 


Rule.  — Measure  the  radius  AG; 
then  multiply  the  square  of  the  radius 
by  3.1^16  (Mens.,  Art.  105). 


Fig.  54. 


To  find  the  area  of  a circular  piece  of  land,  of  which  the 
diameter  is  25  ch.  Ans.  49  A.  0 R.  14  P. 


96.  To  find  the  contents  of  a piece  of  ground  in  the  form  of 

an  ellipse. 


Rule.-  Measure  the  semi-axes  AE, 
GE.  Then  multiply  them  together,  and 
th  eir  product  by  3.  LJfU>. 


SEC.  II.] 


AREA  OR  CONTENTS  OF  GROUND. 


69 


To  find  the  area  of  an  elliptical  piece  of  ground,  of  which 
the  transverse  axis  is  16.08  ch.,  and  the  conjugate  axis  9.72  ch. 

Ans.  12  A.  1 R.  4 P. 

Note  1. — The  following  is  the  manner  of  tracing  an  ellipse 
on  the  ground,  when  the  two  axes  are  known. 

From  C , one  of  the  extremities  of  the  conjugate  axis  as  a 
centre,  and  AE,  half  the  transverse  axis,  as  a radius,  describe 
the  arc  of  a circle  cutting  AE  in  the  two  points  F and  G ; 
these  poins  are  called  the  foci  of  the  ellipse. 

Then,  take  a tape,  the  length  of  which  is  equal  to  AB , 
and  fasten  the  two  ends,  one  at  the  focus  F,  the  other  at  the 
focus  G.  Place  a pin  against  the  tape  and  move  it  around, 
keeping  the  tape  tightly  stretched  : the  extremity  of  the  pin 
will  trace  the  curve  of  the  ellipse. 

Note  2. — In  determining  the  contents  of  ground,  in  the 
examples  which  have  been  given,  the  linear  dimensions  have 
been  taken  in  chains  and  decimals  of  a chain. 

If  the  linear  dimensions  were  taken  in  terms  of  any  other 
unit,  they  may  be  readily  reduced  to  chains.  For,  a chain  is 
equal  to  4 rods,  equal  to  22  yards,  equal  to  66  feet.  Hence, 

1st.  Rods  may  be  reduced  to  chains  and  the  decimal  of  a 
chain , by  dividing  by  Jf. 

2d.  Yards  may  be  reduced  to  chains  and  the  decimal  of  a 
chain , by  dividing  by  22. 

3d.  Feet  may  be  reduced  to  chains  and  the  decimal  of  a 
chain , by  dividing  by  66. 

Note  3. — If  it  is  thought  best  to  calculate  the  area,  with- 
out reducing  the  linear  dimensions  to  chains,  the  result  can 
be  reduced  to  acres* 


70  ELEMENTS  OF  SURVEYING.  [HOOK  II. 

1st.  By  dividing  it  by  160 , when  it  is  in  square  rods 
(Art.  85). 

2d.  By  dividing  it  by  IfSJfO,  when  it  is  in  square  yards 
(Art.  87). 

3d.  By  dividing  it  by  43560,  when  it  is  in  square  feet 
(Art.  87.) 


BOOK  III. 

COMPASS  SURVEYING. 


SECTION  I. 

DEFINITIONS. 

97.  The  Axis  of  the  earth  is  the  immovable  diameter  about 
which  it  revolves ; and  the  poles  are  the  points  in  which  the 
axis  meets  the  surface. 

98.  Any  plane  passing  through  the  axis  of  the  earth  is  called 
a meridian  plane ; and  its  intersection  with  the  surface  is  called 
a meridian  line , or  simply  a meridian. 

99.  All  the  meridians  converge  towards  the  poles,  but  they 
vary  so  little  from  parallelism,  within  the  narrow  limits  of 
surveys  made  with  the  compass,  that  they  may,  without  sensible 
error,  be  regarded  as  parallel  straight  lines. 

100.  If  a magnetic  needle  be  suspended  freely,  and  allowed 
to  settle  to  a state  of  rest,  a vertical  plane  passed  through  its 
axis  is  called  the  p lane  of  the  magnetic  meridian  ; and  its  inter- 
section with  the  surface  of  the  earth  is  called  the  magnetic 
meridian , or  sometimes,  a North  and  South  line.  A line  per- 
pendicular to  a North  and  South  line,  is  called  an  East  and 
West  line. 

101.  A 1 ine  traced,  or  measured  on  the  ground,  is  called  a 
Course ; and  the  angle  which  this  line  makes  with  the  magnetic 


ELEMENTS  OF  SURVEYING. 


[HOOK  III 


meridian,  passing  through  the  point  of  beginning,  is  called  the 
Bearing. 

Thus,  if  we  start  from  the  point 
A,  and  measure  in  the  direction  AB , 
the  line  AB  is  the  course,  and  the 
angle  NAB  is  the  bearing. 

When  the  course,  like  AB,  falls 
between  the  north  and  cast  points, 
and  makes  an  angle  of  40°  with  the 
meridian,  the  bearing  is  read,  north 
46°  east,  and  is  written,  N.  46°  E. 

When  the  course,  like  AC,  falls  between  the  north  and  west 
points,  and  makes  with  the  meridian  an  angle  of  30°,  the  bearing 
is  read,  north  30°  west,  and  is  written,  N.  30°  W. 

When  the  course,  like  AD,  falls  between  the  south  and  west 
points,  and  makes  an  angle  with  the  meridian  of  70°,  the  bearing 
is  read,  south  70°  west,  and  is  written,  S.  70°  W. 

When  the  course,  like  AF,  falls  between  the  south  and  east 
points,  and  makes  with  the  meridian  an  angle  of  70°,  the  bearing 
is  read,  south  70°  east,  and  is  written,  S.  70°  E. 

A course  which  runs  due  north,  or  due  south,  is  designated 
by  the  letter  N,  or  S ; and  one  which  runs  due  east,  or  due  west, 
by  the  letter  E,  or  W. 

102.  If,  after  having  passed  over  a course,  the  bearing  is 
taken  to  the  back  station,  this  bearing  is  called  the  back  sight,  or 
reverse  bearing. 

103.  The  perpendicular  distance  between  the  east  and  west 
lines,  drawn  through  the  extremities  of  a course  is  called  the 
northing  or  southing,  according  as  the  course  is  run  towards  the 
north  or  south.  This  distance  is  also  called  the  difference  of 
latitude,  or  simply  the  latitude,  because  it  shows  the  distance 
which  one  end  of  the  course  is  north  or  south  of  the  other. 


SEC.  I.J 


DEFINITIONS. 


73 


Thus,  in  running  the  course  from  A 
to  B,  AC  is  the  difference  of  latitude, 
north. 


IN' 


W 


c 

H / 



G- 

----- | 

'•  T 

A 

I 

s 

Fig.  57. 


104.  The  perpendicular  distance  between 
the  meridians  passing  through  the  extremi- 
ties of  a course,  is  called  the  departure  of 
that  course,  and  is  east  or  west,  according 
as  the  course  lies  on  the  east  or  west  side  of  the  meridian  passing 
through  the  point  of  beginning. 

Thus,  in  running  the  course  AB,  CB  is  the  departure,  east. 


105.  It  is  found  convenient,  in  explaining  the  rules  for  sur- 
veying with  the  compass,  to  attribute  to  the  latitudes  and 
departures  the  algebraic  signs,  + and  — . 

We  shall,  therefore,  consider  every  northing  as  affected  with 
the  sign  +,  and  every  southing  as  affected  with  the  sign  — . 
We  shall  also  consider  every  easting  as  affected  with  the  sign  -f, 
and  every  westing  as  affected  with  the  sign  — . 


106.  The  meridian  distance  of  a point  is  its  perpendicular 
distance  from  any  assumed  meridian.  Thus,  if  the  distance  be 
estimated  from  the  meridian  NS,  BC  will  be  the  meridian  dis- 
tance of  the  point  B. 


107.  The  meridian  distance  of  a line,  is  the  meridian  dis- 
tance of  its  middle  point,  and  is  east  or  west,  according  as  this 
point  lies  on  the  east  or  west  side  of  the  assumed  meridian. 
Thus,  FG  drawn  through  the  middle  point  of  AB,  is  the 
meridian  distance  of  the  line  AB. 

The  sign  -f  will  always  be  given  to  the  meridian  distance 
of  a point  or  line,  when  it  lies  on  the  east  of  the  assumed 
meridian,  and  the  sign  — , when  it  lies  on  the  west. 


74 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


SECTION  II. 


SURVEYOR’S  COMPASS. 

108.  The  Surveyor’s  Compass  consists  of  a compass-box, 
DCE\  a magnetic  needle;  a brass  plate,  AB , from  twelve  to 
fourteen  inches  long;  two  plane  sights,  AF and  BG  ; two  spirit- 


Fig.  58. 


levels  placed  at  right  angles  to  each  other  ; a brass  head,  K , to  fit 
the  compass  to  a stand,  which  is  sometimes  a tripod  and  some- 
times a single  staff,  called  Jacob-staff,  pointed  with  iron  at  the 
lower  end  so  that  it  may  be  placed  firmly  in  the  ground. 

109.  The  compass-box  DOE  is  circular,  and  generally  about 
six  inches  in  diameter.  At  the  centre  is  a small  pin,  on  which 
the  magnetic  needle  is  poised.  This  needle,  if  allowed  to  turn 
freely  around  the  point  of  support,  will  settle  to  a state  of  rest ; 


SEC.  II.  j SURVEYOR’S  COMPASS.  75 

the  direction  which  it  then  indicates,  is  that  of  the  magnetic 
meridian. 

In  the  interior  of  the  compass-box,  there  is  a graduated 
circle  divided  to  degrees,  and  half  degrees ; the  degrees  are 
numbered  from  the  extremities  of  the  diameter  NS,  both  ways 
to  90°. 

The  length  of  the  magnetic  needle  is  a little  less  than  the 
diameter  of  the  graduated  circle,  so  that  the  needle  can  move 
freely  around  its  centre,  within  the  circle,  and  its  positions  be 
noted  on  the  graduated  arc. 

110.  The  open  sights,  AF  and  BG,  are  placed  at  right 
angles  to  the  plate  AB , and  fastened  to  it  firmly  by  screws. 
The  sights  have  fine  slits  cut  through  nearly  their  whole 
length,  interrupted  at  intervals  by  large  circular  apertures 
through  which  the  object  sighted  upon  is  more  readily  found,  as 
shown  in  the  figure. 

111.  The  spirit-level  is  a small  glass  tube,  slightly  curved 
toward  the  middle,  nearly  filled  with  alcohol,  leaving  a bubble  of 
air  in  the  tube,  and  closed  at  both  ends.  When  the  level  is  in  a 
truly  horizontal  position,  the  bubble  of  air  rests  in  the  middle  of 
the  tube ; when  the  level  is  not  horizontal,  the  bubble  seeks  the 
more  elevated  end.  When  the  two  levels  have  each  the  bubble 
at  the  middle,  each  is  truly  horizontal  and  their  plane  is  hori- 
zontal ; therefore,  the  plane  of  the  graduated  circle  and  the 
magnetic  needle,  which  is  parallel  to  the  plane  of  the  levels,  is, 
also,  horizontal. 

112.  The  brass-head,  by  which  the  compass  is  attached  to 
the  staff,  is  furnished  with  a ball-and-socket  joint  to  give  a 
universal  motion  for  purpose  of  leveling.  Sometimes  a tripod 
is  used  as  a support  instead  of  a staff,  in  which  case  a plumb-bob 
is  attached  immediately  under  the  centre  of  the  graduated  circle 
for  the  purpose  of  accurately  placing  the  instrument  over  any 


76 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


desired  point.  When  a tripod  is  used,  a tripod-head  with  level- 
ing screws  (to  be  described  under  the  transit),  instead  of  a ball- 
and-socket  joint,  is  often  used  for  leveling.  In  the  description 
which  follows,  a ball-and-socket  joint  will  be  assumed. 

113.  To  find  the  bearing  of  any 
course  by  the  compass.  Let  PQ  be  the 
course  whose  bearing  is  desired.  Place 
the  compass  exactly  over  the  point  P, 
by  inserting  the  staff  in  the  ground  at 
that  point,  or  by  the  plumb-bob  if  a 
tripod  is  used.  It  is  usual  for  a sur- 
veyor to  keep  the  south  end  of  the 
compass  towards  him  and  to  read  the 
bearings  from  the  north  end,  and  this 
position  of  the  compass  will  be  assumed 
in  description,  unless  otherwise  stated. 

Turn  the  sights  toward  the  staff  or  object  at  Q.  Bring  the 
bubbles  to  the  middle  of  the  spirit-levels  by  the  pressure  of  the 
hand  on  different  parts  of  the  plate.  Look  through  the  small  slit 
in  the  sight  next  the  person,  and  turn  the  compass  till  the  small 
slit  in  the  sight  opposite  bisects  the  object  at  Q,  being  careful  to 
keep  the  bubbles  at  the  middle  of  the  levels.  Let  the  needle 
come  to  rest  and  take  the  reading  indicated  by  the  north  end  of 
the  needle — it  will  be  the  angle  NPQ , the  bearing.  When  the 
needle  is  at  rest,  it  lies  in  the  magnetic  meridian.  The  line  of 
sights  lies  in  the  direction  of  the  course  ; when  this  line  lies  east 
of  the  meridian,  the  bearing  is  east ; when  it  lies  west  of  the 
meridian,  the  bearing  is  west.  When  the  bearing  is  east , the 
needle  lies  to  the  west  of  the  line  of  sights,  and  the  reverse; 
hence,  to  facilitate  the  reading  the  E and  W letters  on  the  face 
of  the  compass  are  reversed  from  their  natural  position. 

In  order  to  prevent  a merely  mechanical  reading,  the  cover  of 


SEC.  II.] 


surveyor’s  compass. 


77 


the  compass-box  should  be  unscrewed  and  a circle  of  paper 
should  be  fitted  into  the  bottom  of  the  box  so  as  to  conceal  the 
letters;  the  student  will  then  learn  to  read  from  the  needle 
alone,  the  north  end  of  which  bears  a distinguishing  mark  or 
color.  It  is  only  by  such  practice  that  blundering  readings  can 
be  avoided. 

114.  To  find  by  the  compass  the  angle  subtended  at  any 
point  by  two  objects,  take  the  bearing  of  the  course  to  each 
object,  as  explained  in  the  last  article.  The  question  then  is  to 
find  the  angle  between  any  two  courses, 
when  their  bearings  are  known,  which 
may  be  done  as  follows : 

Let  NS  be  a meridian  passing 
through  A. 

Let  AB,  AC,  AH,  AD,  and  AF, 
be  five  courses  running  from  A.  We 
readily  deduce  the  following 


PRINCIPLES. 


AC  is  N 26°  W 
AH  is  N 65°  W 
CAH=  39° 


When  the  meridional  letters 
are  alike,  and  those  of  departure 
► also  alike,  the  difference  of  the 
hearings  is  the  angle  between  the 
courses. 


AB  is  N 46°  E 
AC  is  N 26°  W 
CAB  = 72° 


When  the  meridional  letters 
are  alike,  and  those  of  departure 
unlike,  the  sum  of  the  hearings  is 
J the  angle  between  the  courses. 


78 


ELEMENTS  OF  SURVEYING. 


[BOOK  iil 


AC  is  N 26°  W 
AD  is  S 66°  W 
CAD  = 180°— 92°  = 88° 


When  the  meridional  letters 
are  unlike,  and  those  of  departure 
alike,  the  angle  between  the  courses 
is  equal  to  180°,  minus  the  sum  of 
the  bearings. 


is  N 26°  W 
AF  is  S 66°  E 
CAF  — 180° — 40°  = 140°  , 


When  the  meridional  letters 
are  unlike,  and  those  of  departure 
also  unlike,  the  angle  between  the 
courses  is  equal  to  180 °,  minus  the 
difference  of  the  bearings. 


Note. — The  above  principles  are  deduced,  under  the  sup- 
position that  the  two  courses  are  both  run  from  the  same 
angular  point.  Hence,  if  it  be  required  to  apply  these  rules  to 
two  courses  run  in  the  ordinary  way,  as  we  go  around  the  field, 
the  bearing  of  one  of  them  must  be  reversed  before  the  calcu- 
lation for  the  angle  is  made. 


EXAMPLES. 

1.  The  bearings  of  two  courses,  from  the  same  point,  are 

N 37°  E,  and  S 85°  W ; what  is  the  angle  included  between 
them  ? Ans.  132°. 

2.  The  bearings  of  two  adjacent  courses,  in  going  round  a 

piece  of  land,  are  N 39°  W,  and  S 48°  W;  what  is  the  angle 
included  between  them  ? Ans.  87°. 

3.  The  bearings  of  two  adjacent  courses,  in  going  round  a 

piece  of  land,  are  S 85°  W,  and  N 69°  W ; what  is  the  angle 
included  between  them?  Ans.  154°. 

4.  The  bearings  of  two  adjacent  courses,  in  going  round  a 

piocG  of  land,  are  N 55°  30'  E,  and  S 69°  20'  E ; what  is  the 
angle  included  between  them  ? Ans.  124°  50'. 


SEC.  III.] 


WORK  ON  THE  FIELH. 


79 


SECTION  III. 

WORK  ON  THE  FIELD. 

115.  When  a piece  of  ground  is  to  be  surveyed,  we  begin  at 
some  prominent  corner  of  the  field  and  go  entirely  around  the 
land,  measuring  the  lengths  of  the  bounding  lines  with  the 
chain,  and  taking  their  bearings  with  the  compass.  It  is  not 
material  whether  the  ground  be  kept  on  the  right  haaid  or  on 
the  left,  and  all  the  rules  deduced  for  one  of  the  cases,  are 
equally  applicable  to  the  other.  To  preserve  uniformity,  how- 
ever, in  the  language  of  the  rules,  we  shall  suppose  the  land  to  be 
always  kept  on  the  right  hand  of  the 
surveyor. 

Let  A BCD  be  a piece  of  ground 
to  be  surveyed,  A the  point  where 
the  work  is  to  be  begun,  and  NS  a 
meridian. 

On  a sheet  of  paper,  rule  a single 
column , £ inch  wide , down  the  middle 
of  the  left  hand  page  of  the  note  hook , 
as  in  the  example . 


1ST 


D 


B 


A 

10.00 

A 

7.60 

A 

9.20 

A 

10.40 

A 


(N.  45  E.) 

S.  45J  W. 
(N.  35£  W.) 

S.  36  E. 

(S.  62  W.) 

N.  62  E. 

(S.  31 J E.) 

N.  31i  W. 


80 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


Place  the  compass  at  A,  and  take  the  bearing  to  B, 
which  is  FAB;  suppose  this  angle  has  been  found  to  be  31 1°. 
The  bearing  from  A to  B is  then  N.  31|°  W.  Enter  this  bearing 
in  the  field  notes  at  the  right  of  station  A.  Then  measure  the 
distance  from  A to  B , which  we  will  suppose  to  be  10  ch.  40  1., 
and  insert  that  distance  in  the  column,  above  the  station  mark. 

We  next  take  the  bearing  from  B to  C,  N.  62°  E.,  and  then 
measure  the  distance  BC  = 9 ch.  20  1.,  both  of  which  we  insert 
in  the  notes  as  above. 

At  station  C we  take  the  bearing  to  D,  S.  36°  E.,  and  then 
measure  the  distance  CD  = 7 ch.  GO  1.,  and  place  them  in  the 
notes. 

At  D we  take  the  bearing  to  A,  S.  45 J°  W.,  and  measure 
the  distance  DA  = 10  ch.  We  shall  then  have  made  all  the 
measurements  on  the  field  which  are  necessary  to  determine  the 
contents  of  the  ground. 

116.  The  reverse-bearing  or  back-sight,  from  B to  A,  is  the 
angle  ABH ; and  since  the  meridians  NS  and  HG  are  parallel, 
this  angle  is  equal  to  the  bearing  NAB . The  reverse-bearing 
is,  therefore,  S.  31 E,  and  should  he  entered  in  the  notes  in 
parenthesis  opposite  station  B,  as  in  the  example. 

The  reverse-bearing  from  C,  is  S.  62°  W. ; that  is,  it  is  the 
angle  ICB  — GBC.  And  generally, 

A reverse-hearing , or  hack-sight,  should  always  equal  the  for - 
ward-hearing,  and  differ  from  it  only  in  both  of  the  letters  hy 
which  it  is  designated. 

117.  Tn  taking  the  bearings  with  the  compass,  there  are  two 
sources  of  error.  1st.  The  inaccuracy  of  the  observations;  2d. 
Local  attractions,  or  the  derangement  which  the  needle  expe- 
riences when  brought  into  the  vicinity  of  iron-ore  beds,  or  any 
ferruginous  substances. 


SEC.  IJI.J 


W011K  ON  THE  FIELD. 


81 


To  guard  against  these  sources  of  error,  the  reverse-bearing 
should  be  taken  at  every  station ; if  this  and  the  forward- 
bearing are  of  the  same  value,  the  work  is  probably  right;  but 
if  they  differ  considerably,  they  should  both  be  taken  again. 


Electricity  is  a fruitful  cause  of  annoyance  in  compass  work. 
In  cold,  dry  weather,  any  friction  upon  the  glass  cover  of  the 
needle-box,  even  that  of  a cold,  dry  wind,  charges  it  with  elec- 
tricity and  the  needle  no  longer  traverses  freely.  To  dissipate 
the  charge  touch  the  plate  with  a moistened  linger,  or  breathe 
strongly  upon  it.  If  the  surveyor  uses  a pocket  lens  to  read  the 
needle,  he  must  see  to  it  that  no  iron  or  steel  screws  are  used 
in  the  casing;  nickel-plated  mountings  are  not  admissible; 
hard  rubber  mountings  are  very  troublesome,  as  they  often 
become  highly  charged,  and  will  drag  the  needle  through  90°,  or 
even  180°.  Brass,  or  German  silver,  are  the  most  satisfactory 
mountings. 


118.  In  passing  over  the  course 
AB,  the  northing  is  found  to  be  HB , 
and  the  departure,  which  is  west,  is 
represented  by  AH.  Of  the  course 
BC,  the  northing  is  expressed  by  BG , 
and  the  departure,  which  is  east,  by 
GO.  Of  the  course  CD,  the  southing 
is  expressed  by  Cl,  and  the  departure, 
which  is  east,  by  CF.  Of  the  course 
DA,  the  southing  is  expressed  by  KA, 
and  the  departure,  which  is  west,  by 
DK.  It  is  seen  from  the  figure,  that  the  sum  of  the  northings  is 
equal  to  HB  + BG  = HG ; and  that  the  sum  of  the  southings  is 
equal  to  CI+KA  = PA  = HG;  hence,  the  sum  of  the  northings 
is  equal  to  the  sum  of  the  southings. 

If  we  consider  the  departures,  it  is  apparent  that  the  sum 


29" 


82 


ELEMENTS  OF  SURVEYING. 


[BOOK  III 


of  the  eastings  is  equal  to  GC+  CF  = GF\  and  that  the  sum 
of  the  westings  is  equal  to  AH+DK  = GF;  hence,  the  sum 
of  the  eastings  is  equal  to  the  sum  of  the  westings.  We  there- 
fore see,  that  when  the  survey  is  correct,  the  sum  of  the  northings 
will  he  equal  to  the  sum  of  the  southings , and  the  sum  of  the 
eastings  to  the  sum  of  the  westings. 

It  would,  indeed,  appear  plain,  even  without  a rigorous 
demonstration,  that  after  having  gone  entirely  round  a piece 
of  land,  the  distance  passed  over  in  the  direction  due  north 
must  be  equal  to  that  passed  over  in  the  direction  due  south  ; 
and  that  the  distance  passed  over  in  the  direction  due  east 
must  be  equal  to  that  passed  over  in  the  direction  due  west. 

119.  The  boundaries  of  a field  are  generally  occupied  by 
fences,  and  frequently  also  by  a border  of  shrubbery,  so  that 
chaining  along  the  true  boundary  is  impossible. 

In  such  cases,  it  becomes  necessary  to  measure 
an  offset  at  each  end  of  the  course  (and  at  right 
angles  to  it),  and  of  sufficient  length  to  clear  the 
obstructions ; the  measurement  is  then  made 
between  these  temporary  stations. 

It  is  evident  that  the  bearing  and  length  of 
mn,  the  offset-course,  are  the  same  as  those 
of  MN. 

When  such  offset-courses  are  necessary  for 
several  successive  courses,  errors  are  likely  to  be 
committed,  unless  the  surveyor  is  careful  to  make 
new  offsets  for  each  course. 

To  survey  boundaries  LM,  MN,  NO,  &c.,  along  which  the 
chaining  cannot  he  done,  as  in  Fig.  (54,  proceed  thus : 

Set  the  instrument  at  somo  point,  A for  example,  far  enough 
from  IjM  to  clear  all  obstacles,  usually  five  or  six  feet;  next 


SEC.  III.] 


WORK  OK  THE  FIELD. 


83 


place  a pole  at  B,  make  Bb—Aa  by  measurement,  laying  off  the 
perpendiculars  by  the  eye  alone ; now  sight  B and  note  the  reading. 
While  the  compass  is  sighted  on  B , look  across  two  notches  cut  in 
the  rim  of  the  compass-box  in  a line  perpendicular  to  the  line  of  the 
sights  and  passing  through  the  needle  pivot,  and  fix  a plumb-line 
upon  LM,  as  at  a ; if  the  distance  Aa,  when  re-measured,  should 
differ  much  from  that  used  in  setting  off  Bb , the  offsets  must  be 
corrected  ; now  measure  La  (a  few  links  only),  and,  transferring 
the  link  held  at  a to  A , continue  the  measurement  along  AB  to 
B ; having  set  up  the  compass  at  B,  sight  to  A for  the  back- 
sight, or  reverse-bearing,  and  at  the  same  time  fix  a plumb-line 
at  b by  “ cross-sighting”  as  before  ; transfer  the  link  held  at  B 
to  b,  and  continue  to  M,  and  enter  the  length  of  LM,  thus 
obtained,  in  the  notes. 

Without  moving  the  instrument,  measure  the  perpendicular 
Be,  set  a staff  at  D and  make  Del,  the  perpendicular  to  MN  pro- 
longed, equal  to  Be ; sight  to  D for  the  bearing  of  MN,  cross- 
sight to  fix  c,  and  do  the  same  at  D to  fix  d ; measure  Me,  BD, 
dN',  Me-\-BD — dN  = MN,  which  enter  in  the  notes.  In  like 
manner,  NO  = Ne  + DH+hO. 

120.  It  has  been  customary,  since  the  first  settlement  of  this 
country,  to  use  the  compass  in  all  land  surveys,  so  that  the 
description  of  lands,  in  purchase  and  sale,  and  by  which  they 
are  recognized  in  the  courts,  involves  the  length  and  bearing 
of  each  straight  line  of  the  boundary.  The  method,  therefore, 
is,  at  present,  a necessary  one. 

The  errors  to  which  the  compass  is  liable  are  so  numerous 
and  so  variable,  even  in  the  same  instrument,  that  a change 


84 


ELEMENTS  OF  SUItV  EYING. 


[BOOK  III. 


of  practice  is  very  desirable.  Many  surveyors,  to  insure  a higher 
degree  of  accuracy,  measure  the  angles  of  a field  with  the 
transit,  and  then,  having  determined  the  bearing  of  one  side 
with  sufficient  accuracy,  calculate  the  others  by  a method  to  bo 
shown  in  a subsequent  article. 

121.  In  surveys  of  large  areas,  the  surveying  party  should 
consist  of  at  least  four  persons — viz.,  a compass-man,  a flag- 
man, and  two  chain-men.  In  smaller  areas,  the  work  is  gen- 
erally performed  by  the  surveyor  and  one  assistant;  the  surveyor 
serving  alternately  as  compass-man  and  hind-chainman,  and 
the  assistant  as  flag-man  and  fore-chainman. 

122.  In  recording  the  notes  of  the  survey,  the  advantage  of 
beginning  at  the  bottom  of  the  page  is  this  : that  when  standing 
on  the  line  to  be  surveyed,  and  looking  in  the  direction  we 
propose  to  go,  the  column  in  the  book  lies  before  us  just  as  the 
line  does,  and  all  measurements  made  to  the  right  or  left  of  the 
line  are  recorded  at  the  right  or  left  of  the  column.  In  surveys 
where  many  auxiliary  notes  are  taken,  a diagram  is  an  important 
aid  to  a ready  interpretation  of  the  other  notes. 

GENERAL  EXAMPLE. 

123.  To  explain  the  method,  in  full,  of  making  a compass 
survey  and  recording  the  notes,  we  will  take  an  example  of  a 
farm,  in  which,  in  addition  to  the  usual  survey  of  the  boundary, 
such  other  measurements  are  made  as  to  enable  us  to  make  a 
correct  map  of  the  whole. 

Page  87  represents  a farm  to  be  surveyed,  and  page  86,  the 
notes  which  are  made,  in  the  operations  on  the  field. 

Beginning  with  the  corner  marked  A,  the  bearing  of  the 
line  AB  is  taken.  In  most  cases,  offsets  from  both  A and  B 
would  he  taken,  in  order  that  the  survey  may  be  clear  of  the 
fence,  but  such  offsets  are  not  recorded  ; the  surveyor  must  keep 


SEC.  III.] 


WORK  ON  THE  FIELD. 


85 


m mind  that  it  is  the  boundary  of  the  field  that  is  surveyed,  and 
any  device  by  which  this  is  accomplished  is  no  part  of  his 
notes. 

The  record  of  the  bearing  of  the  first  course  is  entered  at  the 
right  of  the  column  (page  8G),  while  the  letter  designating  the 
station,  is  placed  to  the  left. 

The  symbol  a,  which  signifies  station,  is  placed  in  the 
column,  between  the  letter  and  the  bearing,  for  each  angle  of 
the  farm. 

In  chaining  the  first  course,  the  intersection  of  the  line 
with  any  objects  worthy  of  notice  is  recorded.  The  first  record 
is  of  the  road  leading  to  the  quarry.  As  it  is  an  unimportant 
road,  a single  measurement  of  the  distance  on  the  course  to  its 
centre  is  sufficient  to  locate  it.  The  distance  is  4.30  chains. 

At  11.30  and  12.35  the  sides  of  the  turnpike  are  intersected. 
The  bearing  of  the  road,  at  this  point,  is  also  carefully  taken  and 
recorded. 

The  intersections  of  the  garden  fence  and  of  the  brook  are 
also  noted  (17.40)  and  (18.10) ; and  these,  with  the  entire  length 
of  the  course  (31.95),  close  the  record  of  this  line. 

At  B,  the  back-sight  upon  A is  first  taken,  and  entered  in 
the  notes  opposite  the  ^-station  mark,  as  directed  in  Art.  116. 
The  entry  is  omitted  in  the  example  to  avoid  crowding  and 
confusing  the  notes.  Then  the  bearing  of  BC  is  taken.  Next 
the  bearing  of  the  northernmost  chimney  of  the  farm-house 
is  taken  (N.  73°  E.).  Such  bearings  serve  two  purposes. 
They  aid  in  the  location  of  the  objects  observed,  upon  the  map, 
and  serve  also,  in  case  of  errors,  to  aid  in  detecting  their 
location. 

In  general,  in  surveying  large  or  small  areas,  some  prominent 
point  or  points,  within  the  boundary,  should  be  selected,  and 
their  bearings,  from  different  angles,  carefully  noted. 


Fin.  65. 


MAP  OF  FARM. 


Fig.  66. 


88 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


The  chimney  of  the  farm-house  and  the  oak-tree  in  the 
corner  of  the  wheat-field,  are  thus  employed  in  this  survey. 

At  C,  the  corner  of  the  field,  is  in  the  centre  of  the  brook, 
and  from  this  point  to  D , the  brook  is  the  boundary.  A straight 
line  is  run  between  the  stations,  and  offsets  are  measured  to 
each  bend  of  the  brook. 

It  is  necessary,  in  such  a case,  for  the  chainmcn  to  exercise 
unusual  care  in  keeping  in  the  line  between  the  stations,  other- 
wise the  lengths  of  the  offsets  cannot  be  correctly  measured. 

At  E,  the  bearing  of  the  oak-tree  is  taken  (N.  73^°  E.).  On 
the  course  between  E and  F,  a marsh  is  encountered,  which  the 
chainmen  pass,  by  an  offset  course. 

At  F , another  bearing  is  taken  of  the  oak-tree  (S.  44J°  E. ). 

At  G,  the  bearing  of  the  farm-house  chimney  is  noted 
(S.  26°  E.).  At  G and  H the  bearings  of  the  division-fences  are 
taken.  On  the  course  from  II  to  7,  the  turnpike  is  again 
crossed  ; the  intersection  of  both  sides,  together  with  the  bearing, 
are  carefully  noted. 

From  I to  K,  the  intersection  and  bearing  of  the  fence 
between  the  potato  and  the  wheat  field,  are  recorded.  The  course 
from  K to  A closes  the  survey. 

To  locate  the  buildings  about  the  farm-house,  a few  measure- 
ments would  be  necessary ; but  they  may  begin  with  the  point 
already  located  by  the  bearings  taken  to  the  chimney  nearest  the 
north  end  of  the  house. 

The  dimensions  of  the  buildings,  their  distances  apart,  and 
the  direction  of  one  side  of  each  afford  sufficient  data  for  locating 
them,  correctly,  upon  the  map. 

Note. — The  advantage  of  the  compass  over  other  instruments 
with  which  angles  are  measured,  lies  chiefly  in  this  : that  the 
Bearing  of  a course  may  be  measured  at  any  point  on  the 
line. 


SEC.  III.] 


WORK  ON  THE  FIELD. 


89 


When  the  angle  between  adjacent  sides  is  taken  with  the 
Transit,  the  work  can  only  be  done  at  the  corners  of  the  field  ; 
and  when,  as  frequently  happens,  a hill  intervenes  between  two 
consecutive  stations,  it  becomes  necessary  to  locate  a point  on 
the  hill,  in  the  true  line,  and  then  return  to  the  corner  to 
measure  the  angle ; whereas,  when  the  compass  is  employed,  the 
establishment  of  the  intermediate  point  on  the  hill  affords  the 
means  of  taking  the  proper  bearing  without  going  to  the  angle. 
Furthermore,  the  bearings  may  be  measured  with  the  compass, 
by  placing  it  at  the  alternate  stations  only. 

The  disadvantage  of  these  rapid  methods  is  that  there  is  no 
check  upon  the  needle  readings,  as  there  is  in  back-sighting. 

124.  To  Correct  Local  Attraction.— Suppose  that  bearings 
and  reverse-bearings  have  agreed  for  several  stations,  and  that 
then  a back-sight  differs  from  the  preceding  fore-sight ; it  may 
be  concluded  that  local  attraction  affects  the  needle  at  the  last 
station  only. 

Suppose  the  fore-sight  at  P to  have 
been  N.  30  E.,  and  the  back-sight  at  S 
to  be  S.  28  W. ; suppose  the  fore-sight 
at  8 to  be  N.  2G  W. ; what  is  the  cor- 
rect fore-sight  at  8 ? From  Fig.  67 
we  see  that  the  needle  at  S,  instead  of 
being  parallel  to  the  needle  at  P (as 
indicated  by  the  dotted  needle)  as  it 
should  be,  has  been  deflected  by  local 
attraction  (as  shown  by  the  full  needle), 
so  that  its  south  end  is  moved  2°  to  the  W.,  and  its  north  end  2° 
to  the  E.  of  its  proper  position.  Hence  the  fore-sight  at  S,  being 
taken  from  the  needle  in  a false  position,  is  2°  too  large,  and 
should  be  only  N.  24  W.  By  similar  reasoning,  aided  by  a 
mental  picture  of  the  case,  any  bearing  may  be  corrected. 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


90 


If  no  back-sight  and  its  fore-sight  agree,  then  take  the  mean 
of  the  back-sight  and  fore-sight  which  differ  least;  or  seek  some 
ground  at  a distance,  free  from  local  attraction,  and  then  work 
into  the  plot  to  be  surveyed,  correcting  each  fore-sight  on 
the  way. 


SECTION  IV. 

AREA  OR  CONTENTS  OF  GROUND. 

Having  explained  the  necessary  operations  on  the  field,  we 
shall  now  proceed  to  show  the  manner  of  computing  the  contents 
of  ground. 

125.  The  Traverse  Table  and  its  Uses. — This  table 
shows  the  latitude  and  departure  corresponding  to  bearings  that 
are  expressed  in  degrees  and  quarters  of  a degree,  from  0 to  90°, 
and  for  every  course  from  1 to  100,  computed  to  two  places  of 
decimals. 

The  following  is  the  method  of  deducing  the  formulas  for 
computing  a traverse  table ; by  means  of 
these  formulas  and  a table  of  natural 
sines,  the  latitude  and  departure  of  a 
course  may  be  computed' to  any  desirable 
degree  of  accuracy. 

Let  AD  be  any  course,  and  NAD  its 
bearing;  then  AE  is  the  latitude , and  ED 
the  departure  of  the  course  AD,  to  the  bear- 
ing NAD.  From  Art.  36,  we  have  (using 
nat.  sines) 

AD  : AE  ::  1 : cos  NAD. 

Whence,  lat,  = course  x cosine  of  bearing. 


Fig. 


SEC.  IY.J  AREA  OR  CONTENTS  OF  GROUND.  91 

And  from  Art.  32,  we  have  (since  sine  AED  = 1), 

AD  : ED  : : 1 : sine  NAD. 

Whence,  dep.  = course  x sine  of  bearing. 

We  have  then  the  following  practical  rule  for  computing 
the  latitude  and  departure  of  any  course. 

Loolc  in  a table  of  natural  sines  for  the  cosine  and  sine  of  the 
bearing.  Multiply  each  by  the  length  of  the  course , and  the  first 
product  will  be  the  latitude , and  the  second  will  be  the  departure 
of  the  given  course. 

EXAMPLES. 

1.  The  bearing  is  65°  39',  the  course  69.41  chains:  what  is 


the  latitude,  and  what  the  departure  ? 

Natural  cosine  of  65°  39' 41231 

Length  of  the  course 69.41 

Product,  which  is  the  Dif.  of  Lat.  . . 28.6184371 

Natural  sine  of  65°  39' 91104 

Length  of  the  course 69.41 

Product,  which  is  the  Departure  . . . 63.2352864 


2.  The  bearing  is  75°  47',  the  course  89.75  chains ; what  is 


the  latitude,  and  what  the  departure? 

Natural  cosine  of  75°  47' 24559 

Length  of  course 89.75 

Product,  which  is  the  Dif.  of  Lat.  . . 22.0417025 

Natural  sine  of  75°  47  96937 

Length  of  course 89.75 

Product,  which  is  the  Departure  . . . 87.0009575 


In  this  manner,  the  traverse  table  given  at  the  end  of  the 
book,  has  been  computed.  When  the  bearing  is  given  in  degrees 


92 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


and  quarters  of  a degree,  and  the  difference  of  latitude  and 
departure  are  required  to  only  two  places  of  decimals,  they  may 
be  taken  directly  from  the  traverse  table. 

When  the  bearing  is  less  than  45°,  the  angle  will  be  found 
at  the  top  of  the  page ; when  greater,  at  the  bottom.  When  the 
distance  is  less  than  50,  it  will  be  found  in  the  column 
“ distance,”  on  the  left-hand  page  ; when  greater  than  50,  in 
the  corresponding  column  of  the  right- band  page. 

126.  The  latitudes  or  departures  of 
courses  of  different  lengths,  but  which  have 
the  same  bearing,  are  proportional  to  the 
lengths  of  the  courses.  Thus,  in  the  figure, 
the  latitudes  AG,  AC,  or  the  departures 
GF,  CB,  are  to  each  other  as  the  courses 
AF,  AB. 

Therefore,  when  the  distance  is  greater  than  100,  it  may  be 
divided  by  any  number  which  will  give  an  exact  quotient  less 
than  100;  then  the  latitude  and  departure  of  the  quotient  being 
found  and  multiplied  by  the  divisor,  the  products  will  be  the 
latitude  and  departure  of  the  whole  course.  It  is  also  plain, 
that  the  latitude  or  departure  of  two  or  more  courses,  having 
the  same  bearing,  is  equal  to  the  sum  of  the  latitudes  or  depar- 
tures of  the  courses  taken  separately. 

It  is  always  better  to  obtain  the  lat.  and  dep.  by  addition 
than  by  multiplication;  thus,  if  we  sought  the  lat.  and  dep.  of  a 
course  of  190  ft.,  bearing  36°,  they  would  be  found  by  multiplica- 
tion thus, 

lat.  19  x 10  = 15.37  x 10  = 153.70 
dep.  19  x 10  = 11.17  x 10  = 111.70 

giving  results  containing  ten  times  the  error  of  lat.  and  dep.  of 
19,  as  given  in  the  table.  By  addition  we  should  have 


W 


- M 


E 


S 

Fig.  69. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


93 


lat.  100  + lat.  90  ==  80.90  + 72.81  = 153.71 
dep.  100  + dep.  90  = 58.78  + 52.90  = 111.68 

which  are  closer  approximations  than  the  former. 

Hence,  we  should  always  make  our  multipliers  as  small  as 
possible. 


EXAMPLES. 


1.  To  find  the  latitude  and  departure  of  614,  to  bearing  29 J°. 


Latitude  of  100  x 6 = 522.24 
Latitude  of  14  — 12.18 

Latitude  of  614  534.42 


Departure  of  100x6  = 295.44 
Departure  of  14  = 6.89 

Departure  of  614  = 302.33 


2.  To  find  the  latitude  and  departure  for  the  bearing  62|°, 
and  the  course  7855  chains. 


Latitude  for  7800  . 3602.00 

Lati  tude  for  55  . 25.40 

Latitude  for  7855  . 3627.40 


Departure  for  7800  . 6919.00 

Departure  for  55  . 48.79 

Departure  for  7855  . 6967.79 


Note. — When  the  distances  are  expressed  in  whole  numbers 
and  decimals,  the  manner  of  finding  the  latitudes  and  departures 
is  still  the  same,  except  in  pointing  off  the  places  for  decimals  ; 
but  this  is  not  difficult,  when  it  is  remembered  that  the  column 
of  distances  in  the  table  may  be  regarded  as  decimals,  by 
simply  removing  the  decimal  point  to  the  left  in  the  other 
columns. 


3.  To  find  the  latitude  and  departure  for  the  bearing  47f°, 
and  the  course  37.57. 


Latitude  for  37.00  . 

. 24.88 

Departure  for  37.00 

27.39 

Latitude  for  .57  . 

.38 

Departure  for  .57 

.42 

Latitude  for  37.57  . 

. 25.26 

Departure  for  37.57 

27.81 

94  ELEMENTS  OF  SURVEYING.  [BOOK  III. 

127.  Balancing  the  Work. — The  field-notes  having  been 
completed,  rule  a new  table,  as  below. 

Then  find,  from  the  traverse  table,  the  latitude  and  departure 
of  each  course,  and  enter  them  in  the  proper  columns  opposite 
the  station. 

Then  add  the  column  of  northings,  and  also  the  column 
of  southings;  the  two  sums  should  be  equal  to  each  other. 
If  they  are  not,  subtract  the  less  from  the  greater  ; the  remainder 
is  called  the  error  in  latitude.  Find  the  error  in  departure  in 
the  same  way. 

This  error  for  latitude  or  departure  must  be  distributed 
among  the  latitudes  or  departures  of  all  the  courses,  in  propor- 
tion to  the  length  of  each  course,  observing  to  add  the  correction, 
when  applied  to  the  deficient  column,  and  to  subtract  it,  when 
applied  to  the  other. 

This  may  be  illustrated  by  the  example  of  (Art.  115). 


Stations. 

Bearings. 

Dis. 

Dif. 

Lat. 

Dep. 

Balance. 

N. 

+ 

* s. 

E. 

+ 

W. 

Lat. 

Dep, 

A 

N.  3iy  w. 

10.40 

8.87 

5.43 

+ 8.86 

-5.44 

B 

N.  63°  E. 

9.20 

4.32 

8.13 

+ 4.31 

+ 8.12 

C 

S.  36°  E. 

7.60 

6.15 

4.47 

— 6.15 

+ 4.46 

D 

S.  45 W. 

10. 

7.01 

7.13 

— 7.02 

— 7.14 

Sum  of  courses, 

37.20 

13.1!) 

13.16 

12.60 

12.56 

13.16 

12.56 

Error  in  latitude  . . .03  .64  Error  in  departure. 


The  error  in  latitude,  3 links,  is  to  be  distributed  among  the 
northings  and  southings,  in  proportion  to  the  lengths  of  the 
courses  ; a part  to  be  added  to  the  southings,  and  the  remaining 
part  subtracted  from  the  northings.  The  error  in  departure  is 
similarly  distributed  among  the  castings  and  westings.  For  this, 


Sec.  iy.] 


AREA  OR  CONTENTS  OF  GROUND. 


95 


two  new  columns  are  formed,  called,  the  balanced  latitudes  and 
departures ; and  to  these  columns  the  latitudes  and  departures 
are  transferred,  after  the  corrections  have  been  made ; the  north 
latitudes  being  marked  +,  and  the  south  latitudes  — , in  order 
to  distinguish  them  readily,  and  also,  for  convenience  in  the 
calculations  which  follow. 

The  error  of  .03  in  the  latitudes  is  distributed  among  the 
latitudes,  by  subtracting  1 link  from  each  of  the  northings  of 
courses  A and  B,  and  adding  1 link  to  the  southing  of  course  D. 
This  produces  a balance. 

Of  the  error  of  4 links  in  the  departures,  1 link  is  added  to 
each  of  the  departures  west,  and  1 link  subtracted  from  each 
of  the  departures  east.  This  produces  a balance. 

Note. — When  a knowledge  of  the  conditions  under  which 
the  survey  was  made,  enables  us  to  determine  that  errors  were 
more  likely  to  occur  at  certain  points,  it  is  best  to  apply  the 
corrections  to  those  courses  where  it  seems  probable  the  errors 
were  made. 

128.  The  limit  of  error  to  be  allowed  depends,  of  course, 
upon  the  importance  of  the  survey. 

In  ordinary  farming  districts,  the  error  should  be  as  small 
as  1 link  to  5 or  10  chains  of  perimeter. 

The  “error  of  the  survey”  should  be  considered  as  the  length 
of  the  line  necessary  to  close  the  boundary , and  is  equal  to  the 
square  root  of  the  sum  of  the  squares  of  the  errors  of  latitude 
and  departure.  Thus,  in  the  above  example,  the  error  of  the 
survey  is  5 links.  The  perimeter  being  37.20  chains,  the  error 
is  about  1 link  to  7.45  chains,  or  of  the  perimeter. 

129.  It  will  be  well  to  bear  in  mind  the  fact,  that  if  the 
error  in  the  perimeter  has  been  made  in  one  course  only,  and 
distributed,  by  the  ordinary  methods  of  balancing,  among  all  the 


ELEMENTS  OF  SURVEYING. 


96 


[rook  hi. 


courses,  the  error  in  area  will  be  larger  than  the  error  in 
perimeter. 


130.  When  the  error  is  so  large  that  a re-survey  becomes 
necessary,  the  balancing  should  be  carefully  re-examined,  in 
many  cases,  the  location  of  the  error  may  be  determined 
by  inspection  of  the  computation,  and  a portion  of  the  labor 
of  a re-survey  thereby  saved. 

This  refers  more  particularly  to  those  cases  where  the  error 
is  one  of  chaining,  and  is  mostly  in  one  course.  Errors  of  this 
kind  occur  sometimes  with  experienced  chainmen,  who  draw 
the  chain  properly  between  the  courses,  but  make  occasionally 
an  error  in  counting  the  fractional  part  of  a chain  at  the  end 
of  a course.  In  such  cases,  the  location  of  the  error  may  be 
detected  by  observing,  first  what  columns  contain  errors,  and 
secondly  the  ratio  of  the  errors  of  Latitude  and  Departure. 

When  the  error  in  the  survey  has  been  a single  one,  of  dis- 
tance only,  then  the  ratio  between  the  errors  of  Latitude  and 
Departure  must  be  the  same  as  the  ratio  between  the  Latitude 
and  Departure  of  the  course  to  be  corrected.  If  the  errors  be 
in  northings  and  westings,  then  the  courses  running  either 
North  and  West,  or  South  and  East,  should  be  examined. 


131.  The  surveyor  should  take  every  possible  precaution 
against  errors  in  the  bearings.  This  is  accomplished  by  back- 
sighting,  taking  bearings  of  some  one  object  from  several  sta- 
tions, and  also  by  taking  bearings  of  stations  across  the  field. 
These  precautions  will  give,  in  general,  sufficient  data  for  the 
detection  of  an  error  in  bearing ; for,  by  mapping  the  survey 
ami  drawing  the  lines  to  indicate  the  extra  bearings,  the  error 
is  revealed  by  the  failure  of  the  lines  to  meet  at  a common 
point. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


9? 


132.  One  source  of  error,  m large  surveys  with  the  compass, 
is  frequently  overlooked.  This  is  the  diurnal  variation  ; there 
is  sometimes  as  much  as  15  minutes  variation  during  the  day- 
light hours. 

Errors  from  this  source  can  only  be  avoided  by  testing  the 
compass,  at  intervals  of  two  or  three  hours,  by  taking  the 
bearing  of  the  same  line. 

133.  There  is  one  kind  of  error  frequently  made  in  reading 
the  compass  when  the  bearing  is  nearly  east  or  west.  The 
error  arises  from  reading  North  for  South,  or  the  reverse.  If 
the  survey  is  otherwise  correct,  the  error  in  latitude  is  iust 
twice  ther  latitude  of  the  course  containing  the  error. 

DOUBLE  MERIDIAN  DISTANCES 

134.  After  the  work  has  been  balanced,  the  next  thing  to 
be  done  is  to  calculate  the  double  meridian  distance  of  each 
course. 

For  this  purpose,  any  meridian  line  may  be  assumed.  It  is, 
however,  most  convenient  to  assume  that  meridian  which  passes 
through  the  most  easterly  or  westerly  station  of  the  survey  ; 
and  these  two  stations  are  readily  determined  by  inspecting 
the  field-notes. 

Having  chosen  the  meridian,  let  the  station  through  which 
it  passes  be  called  the  principal  station , and  the  course  which 
begins  at  this  point,  the  first  course.  Care , however,  must  he 
taken  not  to  confound  this  with  the  course  which  begins  at 
station  1,  and  which  is  the  first  course  that  is  entered  in  the 
field-notes. 

It  has  already  been  remarked  (Art.  105),  that  all  departures 
in  the  direction  east  are  considered  as  plus,  and  all  departures 
in  the  direction  west  as  minus. 


98 


ELEMENTS  OF  SURVEY  I NO. 


[HOOK  III 


135.  To  deduce  a rule  for  finding  the  double  meridian  dis- 
tance of  any  course.  Let  SN  be  the  assumed  meridian.  Ix*t 
BC  represent  any  course,  and  AB  the  preceding  course ; also, 
let  D and  E be  their  middle  points.  Draw  EH , DO , and  CM, 
perpendicular  to  the  assumed  meridian 
NS.  Draw  also  Al,  EK,  and  BL,  par- 
allel to  NS.  Then  2 DO  is  the  double 
meridian  distance  of  the  course  BC,  and 
2 Ell  = 2 AT?,  is  the  double  meridian 
distance  of  the  course  AB. 

Now,  2 DO  = ZGK  + 2 KL  + 2 LD; 
but  2KL  = IL  is  the  departure  of  the 
course  AB,  and  2 LD  = MC  is  the  de- 
parture of  the  course  BC;  consequently, 

2 GD  = 2GK+  IL  + MC; 

hence,  the  double  meridian  distance  of  a course  is  equal  to 
the  double  meridian  distance  of  the  preceding  course,  plus  the 
departure  of  that  course,  plus  the  departure  of  the  course  itself : 
if  there  is  no  preceding  course,  the  first  two  terms  become 
zero.  We  therefore  have  the  following 

Rule. — I.  The  double  meridian  distance  of  the  first 
course  is  equal  to  its  departure ; 

II.  The  double  meridian  distance  of  the  second  course 
is  equal  to  the  double  meridian  distance  of  the  first  course, 
plus  its  departure , plus  the  departure  of  the  second  course  ; 

III.  The  double  meridian  distance  of  any  course  is 
equal  to  the  double  meridian  distance  of  the  preceding 
course ; plus  its  departure,  plus  the  departure  of  the 
course  itself. 

Note. — It  should  be  recollected  that  plus  is  here  used  in 
its  algebraic  sense,  and  that,  when  the  double  meridian  distance 


;N 


ii 


Fig.  70. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


99 


of  a course,  and  the  departure  which  is  to  be  added  to  it, 
are  of  different  names,  that  is,  one  east  and  the  other  west, 
they  will  have  contrary  algebraic  signs ; hence,  their  algebraic 
sum  will  be  expressed  by  their  numerical  difference,  with  the 
sign  of  the  greater  prefixed. 

If  the  assumed  meridian  cuts  the  enclosure,  the  double 
meridian  distances  estimated  to  the  east  are  plus,  and  those 
on  the  west  must  be  taken  with  the  minus  sign. 

The  double  meridian  distance  of  the  last  course  should  be 
equal  to  the  departure  of  that  course.  A verification  of  the 
work  is  therefore  obtained,  by  comparing  this  double  meridian 
distance  with  that  departure. 

AREA. 


136.  Let  us  resume  the  example 
of  Art.  115.  We  will  first  write  the 
differences  of  latitude  and  the  double 
meridian  distances  of  the  courses, 
in  the  following  table: 


Fig.  71. 


Stations.  ^ 

Dif.  of  Latitude. 

D.  M.  D. 

Area. 

+ 

Area. 

A 

-f  c B 

-f  %ba 

2 cAB 

B 

+ Bs 

+ %qp 

2BsG 

C 

— Dy 

-j-  2nh 

2ms  CD 

D 

-Df 

+ %ed 

2cmDA 

It  is  evident,  that  cB  multiplied  by  2ba,  or  cA , will  give 
double  the  area  of  the  triangle  cAB.  But  cB  and  ba  are  both 


100 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


plus ; hence,  the  product  will  be  plus,  and  must  be  put  in 
the  column  of  plus  areas.  Double  the  area  of  the  triangle 
BsG,  is  equal  to  Bs  multiplied  by  %qp,  which  product  is  also 
plus. 

The  area  of  the  trapezoid  msCD  is  equal  to  yD,  or  ms,  multi- 
plied by  nh  (Geom.,  Bk.  JV,  Prop.  VII)  ; hence,  double  the 
area  is  equal  to  yD  into  2nh.  But  since  yD  (being  a southing) 
is  minus,  and  2 nh  plus,  it  follows  that  the  product  will  be 
negative;  hence,  it  must  be  placed  in  the  column  of  negative 
areas. 

Double  the  area  of  the  trapezoid  cADm , is  equal  to  Df,  or  me, 
multiplied  by  2 de ; but,  since  Df  is  negative  and  2 de  positive, 
4 the  product  will  be  negative. 

It  is  now  evident  that  the  difference  between  the  two  columns 
is  equal  to  twice  the  contents  of  the  figure  ABCD ; and  since 
the  same  may  be  shown  for  any  other  figure,  we  have,  for  finding 
the  areas,  the  following  general 

Rule.— I.  Multiply  the  double  meridian  distance  of 
each  course  by  its  northing  or  southing,  observing  that  like 
signs  in  the  multiplicand  and  multiplier  give  plus  in  the 
product,  and  that  unlike  signs  give  minus  in  the  product. 

II.  Place  all  the  products  which  have  a plus  sign  in 
one  column,  and  all  the  products  which  have  a minus 
sign  in  another. 

III.  Add  the  columns  separately  and  take  the  dif- 
ference of  their  sums ; this  difference  will  be  double  the 
area  of  the  land. 

Note.— When  offsets  are  measured,  the  figures  of  which 
the  offsets  arc  bounding  lines  are,  practically,  triangles  and 
trapezoids,  and  the  areas  of  these  are  to  be  separately  computed, 
and  added  to  or  subtracted  from,  as  the  case  may  be,  the  area 
obtained  by  the  foregoing  rule. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


101 


137.  We  will  now  make  the  calculations  of  this  example,  in 
numbers,  from  the  field-notes,  which  are  the  following : 


Stations. 

Bearings. 

Distances. 

Diff.  Lat. 

Dep. 

D.  M.  D. 

A 

N 31£°  W 

10.40 

+ 8.86 

-5.44 

+ 18.02 

— 7.14 

- 5.44 

+ 5.44 

*B 

N 62°  E 

9.20 

+ 4.31 

+ 8.12 

+ 8.12 

0 

S 36°  E 

7.60 

-6.15 

+ 4.46 

+ 8.12 
+ 4.46 

+ 20.70 

D 

S 45£°  W 

10. 

1 

© 

-7.14 

4.46 

-7.14 

+ 18.02 

We  see,  from  inspecting  the  notes,  that  B is  the  most 
westerly,  and  D the  most  easterly  station.  Either  of  them  may, 
therefore,  be  taken  for  the  principal  station.  Let  us  assume  B 
for  the  principal  station,  through  which  the  assumed  meridian 
passes,  and  distinguish  it  by  a star,  thus  *. 

Having  done  so,  we  enter  the  departure  8.12  in  the  column 
of  double  meridian  distances,  which  is  the  double  meridian 
distance  of  the  course  from  B to  C.  The  double  meridian 
distances  of  the  other  courses  are  calculated  according  to  the 
rule  ; and  as  the  last,  which  is  that  of  the  course  from  A to 
B , is  equal  to  the  departure  of  that  course,  the  work  is  known 
to  be  right. 

Let  us  now  form  a new  table,  which  will  complete  the 
arithmetical  part  of  the  work. 


102 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


Sta. 

Bearings. 

Dist. 

Dif.  Lat. 

D.  M.  D. 

Area. 

+ 

Area. 

A 

N 31£°  W 

10.40 

+ 8.86 

+ 5.44 

48.1984 

*B 

N 62°  E 

9.20 

+ 4.31 

+ 8.12 

34.9972 

C 

S 36°  E 

9.60 

— 6.15 

+ 20.70 

127.3050 

D 

1 

S 45|°  W 

10. 

— 7.02 

+ 18.02 

126.5004 

83.1956 

253.8054 

83.1956 

# 

2 ) 170.6098 

Area  in  sq.  ch 85.3049 

Ans.  8 A.  2 R.  4.88  P. 


PLOTTING. 


138.  To  make  a plot  of  the  N 
ground,  draw  any  line,  as  NS,  to 
represent  the  meridian  passing 
through  the  principal  station  ; and 
on  this  line  take  any  point,  as  B, 
to  represent  that  station.  Beginning 
at  B , take  the  algebraic  sum  of  the 
balanced  latitudes  and,  also,  of  the 
balanced  departures  of  the  stations 
following.  These  sums  are,  respec- 
tively, the  total  latitudes  and  total  Fro-  72- 

departures  (or  longitudes,  see  Art.  148,  Note)  of  the  several 
courses  with  respect  to  the  principal 
station,  B.  Tabulate  the  results  (see 
example,  p.  94)  as  shown  in  the  margin. 

Through  B,  draw  BX  (Fig.  72)  per- 
pendicular to  NS.  On  NS  lay  off 
(upward  since  4.31  is  plus)  Bs  equal 
to  4.31  units  of  the  scale  on  which  the 


Sta. 

Total  lat. 
from  B. 

Total  dep. 
from  B. 

*b  i 

0.00 

0.00 

0 

4.31 

8.12 

1) 

-1.84 

12.58 

A 

-8.86 

5.44 

SEC.  IV.] 


AREA  OR  CONTENTS  GROUND. 


103 


plot  is  to  be  made;  and  on  BX  lay  off  (to  the  right  since 
8.12  is  plus)  Bf  equal  to  8.12  units  of  the  scale.  From  s,  as  a 
centre,  with  Bf,  or  8.12,  as  a radius,  describe  an  arc  ; and 
from  /,  as  a centre,  with  Bs,  or  4.31,  as  a radius,  describe 
an  arc ; the  intersection  of  these  arcs  will  be  the  position  on 
the  paper  of  Station  C.  On  NS  lay  off  ( downward  since  1.84 
is  minus)  Br  equal  to  1.84  units;  and  on  BX  lay  off  to  the 
right  Bq  equal  to  12.58  units.  From  r,  as  a centre,  with 
Bq,  or  12.58,  as  a radius,  describe  an  arc ; and  from  q,  as  a 
centre,  with  Br,  or  —1.84,  as  a radius,  describe  an  arc;  the 
intersection  of  these  arcs  will  be  the  position  on  the  paper  of 
Station  D.  Determine,  in  like  manner,  the  position  on  the  paper 
of  each  successive  station  ; connect  the  points  so  determined  by 
straight  lines,  and  a complete  plot  of  the  ground  will  be  obtained. 
As  a check,  the  distances  BC,  CD,  &c.,  should  be  measured  on 
the  plot. 

139.  It  is  convenient,  but  not  necessary,  to  take  the  most 
easterly  or  most  westerly  station  as  the  principal  station.  It 
is  simply  to  be  remembered  that  north  latitudes  and  east 
departures  are  plus,  and  south  latitudes  and  west  departures  are 
minus ; that  total  latitudes  and  total  departures  are  algebraic 
sums ; that  plus  latitudes  are  to  be  laid  off  on  the  meridian,  from 
the  principal  station,  upward,  and  minus  latitudes  downivard ; 
that  plus  departures  are  to  be  laid  off  on  the  east  and  west 
line,  from  the  principal  station,  to  the  right,  and  minus  depar- 
tures to  the  left. 

There  are,  of  course,  other  methods  of  plotting;  but  the 
one  here  given  is,  perhaps,  the  best  in  ease  and  accuracy  of 
execution.  If  the  latitudes  and  departures  have  been  correctly 
balanced,  and  the  drawing  carefully  made,  the  survey  will 
certainly  “close.” 


104 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


140.  — E X A M P L E S . 


1.  It  is  required  to  determine  the  contents  and  plot  of  a piece 
of  land,  of  which  the  following  are  the  field-notes,  viz.: 


Sta- 

tions. 

1 

Bearings.  Dist. 

Dif. 

Lai. 

Dep. 

Balanced. 

d.m.d 

+ 

Area. 

+ 

Area. 

N 

-+ 

s 

E 

+ 

w 

Lai.- 

Do,.. 

A 

N46J°  W 20.76 

14.29 

15.06 

+ 14.30 

- 15.04 

15.04 

215.0720 

B 

N 51  E 13.80 

8.54 

10.84 

+ 8.55 

+ 10.86 

10.86 

92.8530 

C 

E 21.35 

2i.ar> 

+ 21.37 

43.09 

D 

S 56°  E 27.60 

15.44 

22.88 

—15.43 

+ 22.90 

87.36 

1347.9648 

E 

S33i°W  18.80 

15.72 

10.31 

—15.71 

-10.29 

99.97 

1570.5287 

F 

N74i°W  30.95 

8.27 

29.83 

+ 8.29 

— 29.80 

59.88 

496.4062 

31.10 

31.16 

55.07 

55.20 

804.3312  2918.49.35 

31.10 

55.07 

804.3312 

Error  . 

. .06 

.13 

Error. 

2)2114.1623 

Am.  105  A.  2 R.  33  P.  1057  08115 


PLOT  OF  THE  GROUND. 


Note. — When  the  bearing  is  due  East  or  due  West,  the  error 
in  latitude  is  nothing,  and  the  corrections  for  latitude  must  be 
distributed  among  the  other  courses.  So,  when  the  bearing  is 
due  North  or  due  South,  the  error  in  departure  is  nothing,  and 
the  error  in  departure  must  be  distributed  among  the  other 
courses.  In  the  examples  for  practice,  we  have  not  been  as 
careful  to  have  as  close  balances  as  must  be  had  in  actual  work 
on  the  field. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


105 


2.  Required  the  contents  and  plot  of  a piece  of  land,  of 
which  the  following  are  the  field-notes. 


Stations. 

Bearings. 

Distances. 

A 

S 34°  W 

3.95  ch. 

B 

S 

4.60 

C 

S 36|°  E 

8.14 

D 

N 59£°  E 

3.72 

E 

N 25°  E 

6.24 

F 

N 16°  W 

3.50 

G 

N 65°  W 

8.20 

Am.  10  A.  0 R.  5 P. 


3.  Required  the  contents  and  plot  of  a piece  of  land,  from 
the  following  field-notes. 


Stations. 

Bearings. 

Distances. 

A 

S 40°  W 

70  rods. 

B 

N 45°  W 

89 

C 

N 36°  E 

125 

D 

N 

54 

E 

S 81°  E 

186 

F 

S 8°  W 

137 

G 

W 

130 

Am.  207  A.  3 R.  33  P. 


10G  ELEMENTS  OF  SURVEYING.  [BOOK  III. 

4.  Inquired  the  contents  and  plot  of  a piece  of  land,  from 
the  following  notes. 


Stations. 

Bearings. 

Distances. 

A 

S 40£°  E 

31.80  ch. 

B 

N 54°  E 

2.08 

C 

N 29^°  E 

2.21 

D 

N 28|°  E 

35.35 

E 

N57°  W 

21.10 

F 

S 47°  W 

31.30 

Ans.  92  A.  3 R.  32  P. 


5.  Required  the  area  of  a survey,  of  which  the  following 
are  the  field-notes. 


Stations. 

Bearings. 

Distances. 

A 

N 42°  E 

5.00  ch. 

B 

East. 

4.00 

C 

N 9°  E 

4.00 

D 

S 69°  E 

5.56 

E 

S 36°  E 

7.00 

F 

S 42°  W 

4.00 

G 

S 75°  W 

10.00 

H 

N 39°  W 

7.50 

If,  in  this  example,  we  assume  A as  the  principal  station, 
the  double  meridian  distances  will  all  be  plus,  and  the  posi- 
tive area  will  exceed  the  negative. 

In  balancing,  we  shall  find  the  error  in  southing  to  be 
.28  ch.,  and  in  westing  .22  eh.  The  area  is  13  A.  0 R.  II  P.  It 
should,  however,  be  remarked,  that  in  all  the  examples  the 
answers  may  be  slightly  varied  by  distributing  the  corrections. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


107 


6.  What  is  the  area  of  a survey  of  which  the  following 
are  the  field-notes?  Make  the  plot. 


Stations. 

Bearings. 

Distances. 

A 

N 75°  00'  E 

54.8  rods. 

B 

N 20°  30'  B 

41.2 

C 

East. 

64.8 

D 

S 33°  30'  W 

141.2 

E 

S 76°  00'  W 

64.0 

F 

North. 

36.0 

G 

S 84°  00'  W 

46.4 

H 

N 53°  15'  W 

46.4 

I 

N 36°  45'  E 

76.8 

J 

N 22°  30'  E 

56.0 

K 

S 76°  45'  E 

48.0 

L 

S 15°  00'  W 

43.4 

M 

S 16°  45'  W 

40.5 

In  this  survey  D is  the  most  easterly  and  I the  most  westerly 
station.  The  area  is  equal  to  110  A.  2 R.  23  P.  It  may  vary  a 
little,  on  account  of  the  way  in  which  the  balancing  is  done. 

7.  What  is  the  area  of  a survey  of  which  the  following 
are  the  notes?  Make  the  plot. 


Stations. 

Bearings. 

Distances. 

A 

S 46J°  E 

80  rods. 

B 

S 51f°  W 

55.20 

C 

West. 

85 

D 

N 56°  W 

110.40 

E 

N 33}°  E 

75.20 

F 

S 74J°  E 

123.80 

Ans.  104  A.  1 R.  1G  P. 


108 


ELEMENTS  OF  SURVEYING.  [BOOK  III. 


8,  Required  the  urea  of  the  farm,  of  which  the  survey 
notes  are  given  in  Art.  123. 


Stations. 

Bearings. 

Distances 
(in  chains). 

A 

N 

08° 

55'  W 

31.95 

B 

N 

8° 

E 

1.40 

C 

S 

87° 

05'  W 

22.89 

D 

N 

30° 

35'  E 

8.39 

E 

N 

4*° 

05'  E 

23.91 

F 

N 

87° 

05'  E 

14.51 

G 

S 

64° 

30'  E 

7.55 

H 

S 

71° 

E 

12.21 

I 

S 

27° 

E 

17.39 

K 

S 

30° 

W 

16.97 

Offsets  between  C and 
D to  be  added, 

15.0450  chains. 

Offsets  to  be  subtract- 
ed, 15.1885  chains. 


9.  To  determine  the  bearing  and  distance  from  one  point  to 
another,  when  they  are  so  situated  that  one  cannot  be  seen 
from  the  other. 

Let  A and  C be  the  two  points,  and  AB  a 
meridian  passing  through  one  of  them.  From 
either  of  them,  as  A,  measure  a course  A 2, 
of  a convenient  length  in  the  direction  toward 
C,  and  take  the  bearing  with  the  compass. 

At  2,  take  the  bearing  of  a second  course,  and 
measure  the  distance  to  3.  At  3,  take  a third 
bearing  and  measure  to  4.  At  4,  take  the  bearing  to  C,  and 
measure  the  distance  from  4 to  C. 

Then,  the  difference  between  the  sum  of  the  northings  and 


Fig.  74. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


109 


the  sum  of  the  southings  will  be  denoted  by  AB  ; and  the 
difference  between  the  sum  of  the  eastings  and  the  sum  of  the 
westings,  by  BC.  The  base  AB,  and  the  perpendicular  BC  of 
the  right-angled  triangle  ABC,  are  then  known.  The  angle 
at  the  base,  BA  C,  is  the  bearing  from  A to  C ; or  the  equal 
alternate  angle  at  C is  the  bearing  from  C to  A,  and  the 
hypothenuse  AC,  is  the  distance. 

Having  measured  the  bearings  and  courses  on  the  field, 
form  a table,  and  find  the  base  and  perpendicular  of  the  right- 
angled  triangle,  in  numbers ; after  which,  find  the  bearing  and 
distance. 


Stations. 

Bearings. 

Distances. 

N. 

s. 

E. 

w. 

A 

N 61°  W 

40  eh. 

19.30 

34.98 

B 

N 42°  W 

41. 

30.47 

27.43 

0 

N 12°  E 

16.10 

15.75 

3.35 

D 

N 47°  E 

32.50 

22.16 

23.77 

AB  = 

= 87.77 

27.12 

62.41 

27.12 

CB  = 35.29  cbu 


To  find  the  angle  BAC,  or  the  bearing  from  A to  C. 

Radius  : tan  A : : AB  : BC, 

or,  AB  : BC  : : R : tan  A ; 

that  is,  applying  logarithms, 

(a.  c.)  log  AB  (87.77) 8.056654 

log  BC  (35.29) 1.547652 

log  R 10. 

log  tan  A 21°  54'  12" 


9.604306 


110 


ELEMENTS  OF  SURVEYING. 


[HOOK  III. 


To  find  the  distance  AC. 


sin  A : R ::  BC  : AC; 

Applying  logarithms, 

( 

(a.  c.)  log  sin  A 21°  54'  12" 0.428242 

log  R 10. 

log  BC  (35.29) 1.547652 

log  AC  94.6  1.975894 


Hence,  the  bearing  and  distance  are  both  found. 

Note. — Had  any  of  the  courses  run  south,  AB  would  have 
been  equal  to  the  sum  of  the  northings,  minus  the  sum  of 
the  southings. 

141.  The  last  problem  affords  an  easy  method  of  finding 
the  bearing  and  length  of  one  of  the  courses  of  a survey,  when 
the  bearings  and  lengths  of  all  the  others  are  known.  It  may 
be  necessary  to  use  this  method  when  there  are  obstacles  which 
prevent  the  measuring  of  a course,  or  when  the  bearing  cannot 
be  taken.  Indeed,  two  omissions  may  in  general  be  supplied  by 
calculation.  It  is  far  better,  if  possible,  to  take  all  the  notes  on 
the  field,  and  it  is  necessary  to  do  so,  if  accuracy  is  required. 
For,  when  any  of  them  are  supplied  by  calculation,  there  are 
no  tests  by  which  the  accuracy  of  the  work  can  be  ascertained, 
and  all  the  errors  of  the  notes  affect  also  the  parts  which  are 
supplied. 

If  necessary,  however,  the  following  omissions  in  the  field- 
notes  may  be  supplied  by  calculation,  viz: 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


Ill 


I.  The  bearing  and  the  length  of  one  course. 

142.  The  following  are  the  field-notes  of  a survey: 


Stations. 

Bearings. 

Distances. 

A 

N 00°  W 

9.72  ch. 

B 

N 17|°  E 

7.65 

C 

N 15J°  W 

9.40 

D 

N (!3|°  E 

10.43 

E 

S 49°  E 

8.12 

F 

S 13|°  E 

8.45 

G 

S 16£°E 

6.44 

H 

With  the  known  bearings  and  distances,  find  the  correspond- 
ing latitudes  and  departures,  as  in  annexed  table : 


Stations. 

Bearings. 

Distances. 

N. 

s. 

E. 

w. 

A 

N 60°  W 

9.72 

4.86 

8.41 

B 

N 17£°  E 

7.65 

7.31 

2.27 

C 

N 15£°  W 

9.40 

9.05 

2.55 

D 

N 63£°  E 

10.43 

4.61 

9.36 

E 

S 49°  E 

8.12 

5.33 

6.13 

F 

S 13£°  E 

8.45 

8.22 

1.98 

a 

S 16}°  E 

6.44 

6.17 

1.86 

H 

25.83 

19.72 

21.60 

10.96 

19.72 

10.96 

Deficiency  p.  1 1 1 A CA  Deficiency 

In  Lat.  South.  O.  11  1U.  K)*±  |n  j)ep  \yeat. 


The  latitude  of  the  course  HA  in  the  plot  (Fig.  75),  must, 


1 12 


ELEMENTS  OF  SUKVEYING. 


[BOOK  III. 


therefore,  be  6.11,  and  its  departure  10.64;  hence,  in  the  right- 
angled  triangle  HP  A, 

PA  = 6.11, 

HP  = 10.64 ; 

The  angle  PAH  = angle  QHA, 
or  the  bearing  of  the  course  HA. 

Find  PAH,  as  in  preceding  ex- 


ample, thus,  B 

AP  : PH  ::  R : tan  PAH ; 
that  is,  by  logarithms,  fiG.  75. 

(a.  c.)  log  AP  (6.11)  = 9.213959 

log  PH  (10.64)  = 1.026942 

log  R — 10. 

log  tan  PAH  = 10.240901 


Hence,  PAH  is  60°  8',  and  since  the  deficiencies  were  in  south 
lat.  and  west  dep.,  the  bearing  of  HA  is  S.  60°  8'  W. 

To  find  the  length  of  the  course,  we  have, 

sin  PAH  : R ::  PH  : HA; 

that  is,  by  logarithms, 

(a.  c.)  log  sin  PAH  {<0 0°  8')  = 0.061887 

log  R = 10. 

log  PH  = 1.026942 

log  HA  = 1.088829 


Hence.  HA  = 12.2 
course.  IIA. 


ch.,  which  is  the  length  of  the  required 


SEC.  IV. ] 


AREA  OR  CONTENTS  OF  GROUND. 


113 


1.  In  a survey  we  have  the  following  notes  : 


Stations. 

Bearings. 

Distances. 

A 

N 31J°  W 

10  ch. 

B 

N 62|°  E 

9.25 

C 

D 

S 45J°  W 

10.40 

What  is  the  bearing  and  distance  from  station  C to  D ? 

( Bearing,  8 38°  52'  E. 
( Distance,  7.03  ch. 


2.  In  a survey  we  have  the  following  notes  : 


Stations. 

Bearings. 

Distances. 

A 

S 40i°  E 

31.80  ch. 

B 

N 54°  E 

2.08 

C 

D 

N 28J°  E 

35.35 

E 

N 57°  W 

21.10 

F 

S 47°  W 

31.30 

What  is  the  bearing  and  distance  from  C to  D ? 

( Bearing,  1ST  34°  47'  E. 

Am.  < 

(Distance,  2. 19  ch. 

II.  The  bearing  of  one  course  and  the  length  of  another,  when 
the  courses  are  (1)  contiguous;  (2)  separated. 

143.  Fi  rst. — Example  when  the  courses  are  contiguous.  In 
Eig.  75,  let  the  bearing  of  DE  and  the  length  of  EF  be  required, 
the  bearings  and  lengths  of  the  other  courses  being  known. 


J&4  ELEMENTS  OP  SURVEYING.  [BOOK  III. 


As  before,  find  latitudes  and  departures  for  known  bearings 
and  distances,  obtaining  annexed  table  : 


Stations. 

Bearings. 

Distances. 

N. 

S'. 

E. 

w. 

A 

N 60°  W 

9.72 

4.86 

8.41 

B 

N17J°  E 

7.65 

7.31 

2.27 

C 

N 15f°  W 

9.40 

9.05 

2.55 

D 

10.43 

E 

S 49°  E 

F 

S 13J°  E 

8.45 

8.22 

1.98 

G 

S 16f°  E 

6.44 

6.17 

1.86 

H 

S 60°  8' W 

12.27 

6.11 

10.64 

Draw  a line  DF  and  find  the  bearing  and  the  length  of  DF 
in  the  plot  ABCDFGH,  as  in  Art.  142.  The  length  will  be 
found  to  be  15.506,  and  the  bearing  87°  20'  19"  ; the  angle  DFF 
is  then  equal  to  87°  20'  19"— 49°  (the  bearing  of  EF)  = 38°  20' 
19"  ; then  in  the  triangle  DEF,  the  sides  DE  (10.43),  and  DF 
(15.506),  and  the  angle  DFE  (38°  20'  19")  are  known,  and  the 
angle  DEF  will  be  found  from  the  proportion, 

DE  : DF  ::  sin  DFE  : sin  DEF 

to  be  equal  to  112°  45' ; hence,  the  bearing  of  DE  is  112°  45'— 49° 
= 63°  45'.  The  length  of  the  course  AT^will  be  found  from  the 
proportion, 

sin  DFE( 38°  20'  19")  : sin  FDE  ( 28°  54'  41")  : : (10.43)  : EF 

to  be  equal  to  8.12. 

Note. — In  finding  the  angle  DEF  doubt  arises  as  to  the 
proper  angle,  corresponding  to  the  logarithmic  sine,  to  be  taken; 
DEF  may  be  either  67°  15',  or  112°  45'  (see  Art.  39);  and  hence 
the  bearing  of  DE  may  be  cither  18°  15,  or  63°  45',  without 


SEC.  IV. j 


AREA  OR  CONTENTS  OE  GROUND. 


115 


affecting  the  length  of  the  given  course  DE.  If  the  smaller 
angle  67°  15'  is  taken  as  value  of  DEF \ then  angle  FDE  is  74° 
24'  41”,  instead  of  28°  54'  41”,  and  the  length  of  the  required  side 
EF,  as  determined  from  the  proportion, 

sin  DFE  : sin  FDE  : : DE  : EF 

will  be  16.19,  instead  of  8.12,  without  affecting  the  given  bearing. 

When  the  length  of  the  side  EF \ whose  bearing  is  required,  is 
not  greater  than  that  of  the  auxiliary  line  DF,  the  method  given 
is  of  no  service,  unless  some  independent  means  exist  of 
determining  which  of  the  two  possible  solutions  is  the  one  to  be 
taken. 


The  following  are  the  notes  of  a survey : 


Stations. 

Bearings. 

Distances. 

A 

North 

12.84  ch. 

B 

N 32°  E' 

17.82 

C 

N 80°  E 

24. 

, D 

S 48°  E 

27. 

E 

S 18°  W 

F 

46.21J 

Required  the  length  of  the  course  EF \ and  the  bearing  of  the 
course  FA. 

Length  of  EF  = 28.60. 

Bearing  of  FA  = N 73°  28'  21”  W. 

144.  Second. — Example  when  the  courses  are  separated.  In 
Fig.  75  let  the  bearing  of  AB  and  the  length  of  DE  be  unknown. 

From  B,  Fig.  76,  one  extremity  of  the  course  whose  bearing 
is  unknown,  draw  BP  parallel  and  equal  to  DE,  whose  length  is 
required.  Draw  then  PQ  and  QE,  equal  and  parallel  to  BC  and 
CD,  respectively ; and  draw  PA.  The  lengths  and  bearings  of 


116 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


the  courses  PQ  and  QE  are  the  same 
as  those  of  BC  and  CD,  respectively ; 
hence,  in  the  figure  APQEFGH  the 
lengths  and  bearings  of  all  the  courses 
except  AP  are  known ; and  the  length 
and  bearing  of  AP  may  be  found  as 
in  Art.  142.  Then  in  the  triangle 
APB,  the  length  of  AP  and  A B are 
known,  and  the  angle  APB  (which  is 
the  sum  or  difference  of  the  bearings 
of  BP  and  AP) ; hence,  as  in  Art.  143, 
the  length  of  BP  or  its  equal  DE,  and  the  bearing  of  AB  may  be 
found.  (Gillespie’s  Land  Surveying,  p.  299).  The  same  doubt 
as  to  the  proper  solution  exists  here,  as  in  the  previous 
example. 

111.  The  bearings  of  two  courses  when  they  are  (1)  contiguous ; 

(2)  separated. 

145.  First. — Example  when  they  are  contiguous.  In  Fig. 
75,  let  the  bearings  of  DE  and  EF  be  unknown.  Draw  a line 
from  D to  F and  find  its  length  and  bearing  from  the  latitudes 
and  departures  of  the  other  courses  as  before.  Then  in  the  triangle 
DEF  the  three  sides  are  known  and  the  three  angles  may  be  found 
as  in  Article  41.  From  the  angle  EDF and  the  bearing  of  DF, 
the  bearing  of  DE  may  be  found ; and  from  the  angle  EFD  and 
the  bearing  of  DF,  the  bearing  of  EF  may  be  found. 

146.  Second. — Example  when  the  courses  are  separated.  Let 
AB  and  DE,  Fig.  75,  be  the  courses  whose  bearings  are  un- 
known. Assume  the  figure  as  in  Fig.  70.  Then  the  bearing 
and  length  of  AP  may  be  found  as  before.  In  the  triangle  APB, 
all  the  sides  are  known  and  the  angles  may  be  found  as  in  the 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


11? 


preceding  article.  Then  the  bearing  of  BP,  or  its  parallel  DE, 
may  be  found  from  the  angle  BP  A and  the  bearing  of  PA  ; and 
the  bearing  of  AB,  from  the  angle  PAB  and  the  bearing  of  AP 
(Gillespie’s  Land  Surveying). 

IV.  The  lengths  of  two  courses. 

147.*  Let  A = bearing  of  the  first 
unmeasured  course,  FG,  of  the  survey. 

B = bearing  of  the  second  course, 

GH. 

C = angle  between  the  courses. 

L — total  latitude,  and 

D = total  departure  of  the  two  un- 
measured courses,  obtained  by  comput- 
ing the  total  latitudes  and  total  departures 
of  the  measured  courses,  HK  to  PF,  as  in  Art.  142. 

Let  x = length  of  the  first  course. 

y = length  of  the  second  course. 

We  shall  then  have, 

L = (lat.  x -plat,  y)  = x cos  A+y  cos  B.  (1) 

D = (dep.  x + dep.  y)  = x sin  A +y  sin  B . (2) 

Multiply  (2)  by  cot  B, 

D cot  B = x sin  A cot  B-\-y  cos  B.  (3) 

Subtract  (3)  from  (1)  and  solve  with  regard  to  x, 

_ L — D cot  B 
cos  A — sin  A cot  B 

Multiply  both  numerator  and  denominator  by  sin  B, 


* Prof.  II.  S.  Munroc.  E.M.,  Ph.D.,  and  J.  Woodbridge  Davis,  C.E.,  Pli.D.,  in  “School  of 
Mines  Quarterly,”  for  November,  1881. 


118 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


_ L sin  B—D  cos  B 
X - sin  (A-B)  5 

or,  substituting  for  sin  ( A—B ) its  value, 


L sin  B—D  cos  B 
sin  C 


In  like  manner  we  have, 

D cos  A — L sin  A 
y ~ sin- O 


(4) 


(5) 


By  taking  into  account  the  signs  of  the  different  quantities, 
the  above  formulas  (4)  and  (5)  will  solve  any  case  that  may 
arise,  not  only  when  the  unmeasured  sides  are  contiguous,  as  in 
the  figure,  but  any  two  sides  of  the  polygon,  e.g.,  sides  FG 
and  ON,  provided  only  that  the  courses  are  not  parallel,  or 
nearly  so,  as  FG  and  MN. 

The  following  table  gives  the  proper  signs  in  a convenient 
form  for  reference: 


Signs  of  L.  and  D. 

Signs  of  functions  of  A.  and  B. 

Lat.  N. ; L.  is  -f- . 
Dep.  E.;  D.  is  +. 
Lat.  S. ; L.  is  — . 
Dep.  W. ; D.  is  — . 

N.E.  bearing;  sin  +,  cos  +. 
S.E.  bearing;  sin  +,  cos  — . 
S.W.  bearing;  sin  — , cos  — . 

N. W.  bearing;  sin  — , cos  4-. 

To  find  value  of  C. 

C = (A  — B)  if  courses  are  in  same  or  in  opposite  quadrants. 
C = (A B)  if  the  courses  are  in  adjacent  north  or  adjacent 
south  quadrants. 

C—  1 80°  — ( A -f-  B)  if  courses  are  in  adjacent  east  quadrants 
or  adjacent  west  quadrants. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


119 


EXAMPLE. 


Sta. 

Bearings. 

Dist. 

Latitude. 

Departure. 

N. 

s. 

E. 

w. 

A 

East. 

48  ch. 

48.00 

B 

N.  71 E. 

5.18 

1.64 

4.91 

0 

S.  13£°  E. 

34. 

33.13 

7.65 

D 

South. 

12. 

12.00 

E 

S.  89°  10'  W.  (A) 

(*) 

F 

N.  80°  20'  E.  (B) 

(y) 

1.64 

45.13 

60.56 

L=  +43.49  D = —60.56 

From  the  above  table  we  find  that,  in  order  to  balance  the 
latitude  and  departure  columns,  the  total  latitude  of  the  two 
unknown  courses  must  be  43.49  North,  and  the  total  departure 
60.56  West.  Substituting  in  the  formulas, 

_ 43. 49  x sin  80°  20'  + 60. 56  x cos  80°  20'  _ 

* ~ sin  8°  50'  ’ 

_ 60.56  x cos  89°  10' + 43. 49  x sin  89°  10' 
y ~ sin  8°  50' 

Solving,  x = 345.41 ; y = 288.91. 

As  a check,  the  latitudes  and  departures  of  the  two  sides 
should  be  computed,  and  the  survey  closed  as  below. 


Stations 

Bearings. 

Dist. 

Latitude. 

Departure. 

N. 

s. 

E. 

w. 

A 

East. 

48. 

48.00 

B 

N.  71J°  E. 

5.18 

1.64 

4.91 

O 

S.  13£°  E. 

34. 

33.13 

7.65 

D 

South. 

12. 

12.00 

E 

S.  89°  10'  W. 

345.41 

5.02 

345.37 

F 

N.  80°  20'  E. 

288.91 

40 

00 

284.81 

50. 15 

50.15 

345.37 

345.37 

120 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


The  following  are  the  field-notes  of  a survey  : 


Stations. 

Bearings. 

Distances. 

A 

North. 

7.81  ch. 

B 

S.  76±°  E. 

C 

S.  10°  47'  W. 

28.42 

D 

£ 

o 

00 

27.12 

E 

N.  4£°  W. 

F 

East. 

16.68 

Required  the  distance  from  B to  C and  from  E to  F. 

B to  Cy  17.87  ; E to  Fy  21.82. 

148.  Another  Method  of  Determining  Areas.* 

The  area  of  any  right-line  figure  may  be  decomposed  into  a 
series  of  right  trapezoids  and  triangles  by  letting  fall  from  the 
vertices  perpendiculars  to  any  fixed  straight  line,  called  the 
Base. 

The  triangles  may  be  considered  trapezoids  having  one 
parallel  side  reduced  to  zero.  Some  of  the  trapezoids  may  be 
subtractive;  but  we  may  say  that  the  algebraic  sum  of  the 
trapezoids  composing  the  series  is  the  area  of  the  figure.  If  the 
figure  be  bounded  by  a continuous  or  a broken  curve,  its  area  is, 
approximately,  equal  to  that  of  a right-line  figure  whose  vertices 
are  contained  in  the  curve.  By  properly  locating  these  vertices, 
we  may  approximate  the  area  as  closely  as  may  be  required. 

Area  of  a Series  of  Right  Trapezoids. 

Before  deducing  the  necessary  rules,  a few  preliminary  defini- 
tions will  be  given. 


* J.  Woodbridge  Davis,  C.E.,  Ph.D.,  1879. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


121 


The  position  of  points  in  a plane  may  be  fixed  by  referring 
them  to  two  intersecting  straight  lines  of  the  plane.  The  two 
intersecting  lines  are  called  Coordinate  Axes. 

Thus,  let  Jl'  and  YY' 
intersect  at  A ; the  position 
of  the  point  P is  known  when 
RP,  its  distance  from  YY'  on 
a line  parallel  to  XX',  and  X 
QP,  its  distance  from  XX  on 
a line  parallel  to  YY',  are 
known.  RP,  or  its  equal  Y 

A Q,  is  called  the  Abscissa  fig-  78. 

of  the  point  P;  and  QP,  or  its  equal  AR,  is  called  the 
Ordinate  of  P.  AQ  and  QP  together  are  called  the  Co- 
ordinates of  P.  The  point  A,  in  which  XX'  and  YY' 
intersect,  is  called  the  Origin  of  Coordinates.  When  XX' 
and  YY'  are  perpendicular  to  each  other,  as  is  most  convenient, 
the  coordinates  of  points  referred  to  them  are  called  Rectangu- 
lar Coordinates. 


Y 


Consider  any  series  of  five 
right  trapezoids  lying  consecutive, 
with  their  bases  on  one  straight 
line.  The  area  of  the  series, 
as  found  by  ordinary  method, 
the  symbols  of  Fig.  79  used,  is 


Fie.  79. 


^Dx(aYi)Y{D,-Dx){bYc)Y{D,-D,){c  + d)Y(D,-Dz) 

0 d+e)  + (D-Dl)(e+f )].  (1) 

This  may  be  transformed  into 


i [D,(a-c)  + D2(b~d)+D,{c-e)  + Di(d-f)  + D(e+f)\.  (2) 
From  this  is  derived  the  following  rule,  which  is  true,  at  least. 


ELEMENTS  OF  SURVEYING. 


122 


[book  III. 


for  any  series  of  five  right  trapezoids  arranged  as  in  Fig.  79. 
For  convenience  let  this  be  called  Rule  A. 


Rule  A. — To  find  the  area  bounded  partly  by  any  broken 
line,  determined  from  a base  line  by  means  of  rectangular 
ordinates,  and  otherwise  bounded  by  the  base  line  and  terminal 
ordinates : 

Multiply  the  distance  of  each  intermediate  ordinate 
from  the  first  by  the  difference  between  the  two  adjacent 
ordinates,  always  subtracting  the  following  from  the  pre- 
ceding in  order  along  the  broken  line.  Also,  multiply 
distance  of  last  ordinate  from  first  by  the  sum  of  last 
two  ordinates.  Divide -the  sum  of  these  products  by  2. 

If  this  rule  apply  to  a series  of  n trapezoids,  its  area  is 

i \D\  (a — c)  + D<i  (b — d ) -f-  etc.  -f-  Dn_x  (i — k)  + Dn  (j  -f-  &)].  (3) 

Add  another  trapezoid  to  the  latter  end  of  this  series.  Its 
area  is 

i (A.+i —Dn)(k  + l).  (4) 

Add  this  to  (3).  The  result  is  the  area  of  a series  of  n + 1 
trapezoids,  and  is  expressed  by  the  following  : 

| [./li  {a — c)  + etc.  -\-Dn_x  ( i — k ) + Dn  (y — V)  + Dn+l  {k-\- 1). 

Therefore,  the  rule  applies  to  a series  of  w-fl  trapezoids. 

If  from  the  series  of  n trapezoids  the  last  trapezoid,  whose 
area  is 

i (/>„-/>„_,)  (j  + k), 

be  subtracted,  the  area  of  the  remaining  series  of  n — 1 trapezoids 
is  the  following : 

| [ /9,  ( a — c)  4*  D'i  {b — d)  + etc.  + Dn_i  (i  +y)J. 

Therefore,  the  rule  applies  to  a series  of  n— 1 trapezoids. 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


123 


It  is  certain  that  the  rule  applies  to  a series  of  five  trapezoids. 
Let  n — 5.  Then  n- f-1  = 6,  and  n — 1 = 4.  Consequently,  the 
rule  applies  to  a series  of  six  trapezoids,  and  to  a series  of  four. 
For  this  reason  it  applies  to  series  of  three  and  seven  trapezoids, 
and  so  on.  Because  a series  containing  any  number  of  trape- 
zoids may  be  found  by  continually  adding,  or  subtracting,  one 
trapezoid  to,  or  from,  a series  of  five,  and  because  the  application 
of  the  rule  is  not  affected  by  each  such  operation,  it  follows 
that  Rule  A applies  to  a series  consisting  of  any  number  of 
trapezoids. 

If  that  portion  of  the  broken  line  which  forms  the  upper  side 
of  the  last  trapezoid  be  perpendicular  to  base,  whether  directed 
towards  it  or  from  it,  the  area  of  the  last  trapezoid  is  zero. 
But  expression  (4),  which  represents  this  area,  is  zero.  There- 
fore the  rule  conforms  to  this  case.  If  the  last  portion  of  broken 
line  be  retrogressive,  as  shown  in 
Fig.  80,  where  the  line  termi- 
nates at  Kz , or  K4,  the  last 
trapezoid  must  be  subtracted  in 
order  that  the  required  area — 
between  broken  line  and  base 
limited  by  terminal  ordinates — may  be  obtained.  But  in  this 
case  expression  (4)  is  intrinsically  negative,  and  therefore  should, 
as  before,  be  algebraically  added  to  the  preceding  area.  Hence 
the  rule  conforms  to  this  case. 


J 

\ 4 

*3 

( 

Fig.  80. 


The  statements  of  last  paragraph  are  true,  whatever  the 
number  of  trapezoids  in  the  series;  even  should  the  trapezoid 
added  compose,  alone,  the  series.  In  consequence,  since  every 
possible  broken  line  is  some  succession  of  progressive,  retro- 
gressive, and  perpendicular  elements,  and  because  the  addition  of 
each  element  and  its  accompanying  trapezoid  does  not  affect  the 
rule,  rule  A,  or  formula  (3),  is  perfectly  general,  whatever  may 
be  the  complication  of  the  broken  line. 


124  ELEMENTS  OF  SURVEYING.  [ROOK  III. 

Example. — Let  it  be  required  to  determine  the  area  of  a tract 
of  land  whose  description  is  as  follows: 

Beginning  at  a poplar  on  the  south  bank  of  the  Cumberland 
river;  thence  south  31  poles  to  a stone  monument;  thence  east 
180  poles  to  a beech  ; thence  north  40  poles  to  a sycamore  on  the 
bank  of  said  river ; thence,  with  the  meanderings  of  the  same,  to 
the  beginning. 

It  was  found  convenient  to  determine  the  irregular  boundary 
by  means  of  offsets  from  the  long  straight  side.  The  positions 
and  lengths  of  these  offsets  are  recorded  in  the  first  two  columns 
below.  To  obtain  the  area  by  Rule  A,  a third  column  of  alternate 
differences  is  obtained  ; and  in  the  fourth  column  are  placed  the 
products  of  corresponding  quantities  in  first  and  third  columns: 
half  the  algebraic  sum  is  the  area. 


Distances. 

Offsets. 

Differences. 

Products. 

0 

31 

15 

30 

— 8 

-120 

45 

39 

-12 

-540 

77 

42 

0 

0 

86 

39 

-2 

-172 

113 

44 

3 

339 

161 

3G 

4 

644 

180 

40 

7G 

13680 

14GG3 
— 832 


2 ) 13831_ 
Square  rods  . . . G915J 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


125 


The  advantages  of  this  method  of  calculation  are  readily  per- 
ceived by  inspection  of  the  above  example,  where  every  product 
except  the  last,  is  found  by  mental  work.  To  illustrate  this 
more  forcibly,  let  us  suppose  the  base  line  to  be  moved,  for 
instance,  2658  poles  directly  south.  Now,  all  the  products 
except  the  last  remain  as  simple,  in  fact  the  same,  as  before, 
while  the  last  only  is  increased. 

The  old  method  of  calculation  would  now  require  every 
product  to  be  increased. 

If,  to  find  the  area  of  an  irregular  figure,  parallel  cross 
measurements  be  made  at  appropriate  places,  intersecting  a given 
base  line  at  known  points,  the  figure  may  be  considered  a series 
of  trapezoids.  If  the  base  line  be  perpendicular  to  the  direction 
of  cross-measurements,  Rule  A obviously  applies.  If  the  base 
line  be  inclined  to  the  cross  lines,  Rule  A may  be  applied, 
but  the  final  result  must  be  multiplied  by  the  sine  of  the  angle  of 
inclination. 


EXAMPLES. 

1.  Wallace  Colyar  agrees  to  sell  to  James  Beckwith  at  ninety 
dollars  per  acre,  a piece  of  unimproved  bottom  land,  lying 
between  the  west  bank  of  the  Delaware  river  and  the  foot  of  the 
adjacent  hill.  The  description  of  the  survey  is  as  follows  : 

Beginning  at  a willow  on  the  west  bank  of  Delaware  river, 
this  being  the  northeast  corner  of  Henry  Gillespie’s  home  place ; 
thence  with  the  meanderings  of  the  bank  of  the  river — [numerous 
courses  here  omitted] — to  a white  oak,  the  corner  of  James  Beck- 
with’s land  ; thence  west  with  his  south  boundary  to  a spring  at 
the  foot  of  the  hill ; thence  meandering  with  the  foot  of  the  hill — 
[numerous  courses  omitted] — to  a black  walnut,  Henry  Gillespie’s 
northwest  corner ; thence,  with  his  line  east  to  the  beginning. 
The  east  and  west  being  sinuous  but  unchangeable  natural  bound- 
aries, it  is  only  required  to  determine,  as  easily  and  accurately  as 


126 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


possible,  the  area.  Accordingly,  a line  is  run  due  north  through 
the  land  and  cross- widths  are  taken  at  right  angles.  The 
distances  on  base  line  and  the  lengths  of  cross  lines  are  recorded 
in  the  columns  below.  It  is  required  to  find  the  area  in  acres 
and  the  sum  in  dollars  to  be  paid  for  the  strips  of  land. 


Distances. 

Widths. 

0 

38  rods. 

19 

29 

40 

32 

78 

36 

111 

51 

145 

45 

173 

40 

200 

44 

228 

50 

254 

55 

290 

58 

337 

43 

Ans.  94.05625  acres ; $8465.06. 

2.  Caleb  Hopkins  employs  John  Bryant  to  cut  and  burn 
the  underbrush  in  an  irregular  piece  of  hilly  woodland,  at 
one  dollar  and  a half  per  acre,  the  agreement  being  that 
the  chain  shall  be  allowed  to  rest  upon  the  surface  through- 
out its  length,  and  not  be  stretched  horizontally. 

A base  line  was  run  N.  20°  W.  through  the  land,  and  cross- 
measurements were  taken  in  directions  due  cast  and  west.  The 
distances  in  base  lino  and  the  lengths  of  cross  lines  are  as 
follows : 


SEC.  IV.] 


AREA  OR  CONTENTS  OF  GROUND. 


12? 


Distances. 

Widths. 

0 

0 rods. 

10 

19 

26 

41 

55 

49 

78 

52 

106 

61 

129 

50 

153 

55 

175 

39 

180 

45 

194 

31 

209 

0 

To  find  the  Area  of  a Polygon. 

It  has  been  proved  that  Rule  A is  true  for  any'  series  of 
trapezoids,  whatever  may  be  the  complication  of  the  broken 
line  which  forms  the  upper  sides  of  the  right  trapezoids  ; there- 
fore the  rule  is  true  when  the  broken  line  returns  to  its  point  of 
beginning.  But  in  this  case  the  area  required  as  the  rule  is  the 
area  enclosed  by  the  broken  line  itself,  or  is  the  area  of  any 
polygon.  Now  D,  the  distance  between  first  and  last  ordinates, 
becomes  zero,  causing  the  only  term, 
which  depends  upon  a sum  of  two 
ordinates  for  one  factor,  to  disappear. 

To  treat  this  case  in  the  most  gen- 
eral way,  suppose  B , G,  D , E , F (Fig. 

81),  to  be  points  anywhere  situated 
in  a plane,  and  referred  in  position  by  rectangular  co-ordinates 
to  any  point  A in  same  plane;  and  suppose  points  to  be  con- 


B 


X> 


.A/ 


Fig.  81. 


128 


ELEMENTS  OF  SURVEYING. 


[BOOK  HI. 


nected  in  the  order  named  by  a broken  line,  ending  again  at  B. 
Let  the  ordinates  of  the  points  be  denoted  by  a,  b,  c,  d,  e,  /,  and 
the  abscissas  by  a',  b\  c',  d' , e' , f . Connect  the  origin  by  a 
straight  line  with  any  of  the  points,  as  B.  Then  the  area  of 
the  polygon  ABCDEFBA , which  is  equal  to  the  area  of  the 
polygon  BCDEFB , is  by  Rule  A, 

i (ci—c)+c'  (b-d)-\-d'  (c— e)  -\-e'  ( d—f ) -f  /'  (e— b) 

+ V (/  — a )]  (5) 

= i L b'  (/  - c)  + d (b  - d)  + d'  (c  - e)  + e'  (d  - b) 

+ f'(e-b)]  (G) 

= ~i  [Hf~  o')  + c {b'-d')  + d ( c'-e' ) + 0 (d’-f) 

From  expressions  (6)  (7)  can  be  framed  the  following  rule  : 

Rule  B. — To  find  the  area  of  any  polygon  whose  vertices  are 
fixed  by  rectangular  co-ordinate  measurements: 

Multiply  the  abscissa  of  each  vertex  by  the  difference 
between  the  ordinates  of  the  two  adjacent  vertices ; or, 
multiply  the  ordinate  of  each  vertex  by  the  difference 
between  the  abscissas  of  the  two  adjacent  vertices ; always 
making  the  subtraction  in  the  same  direction  around 
the  polygon.  Half  the  sum  of  these  products  is  the 
area. 

If  the  origin  be  placed  at  a vertex,  one  term  vanishes  which- 
ever way  the  rule  is  used.  If  one  of  the  co-ordinate  axes  be 
passed  through  two  vertices,  two  terms  vanish,  when  the  rule  is 
used  one  way. 


SEC.  IV.  J AREA  OR  CONTENTS  OF  GROUND.  * 129 

The  determination  of  the  areas  of  polygons  is  a problem  of 
frequent  occurrence  in  all  branches  of  engineering  and  many 
other  professions.  Rule  B is  invariably  simpler  than  the 
ordinary  formula  for  this  case,  and  is  therefore  presented  above 
in  the  most  general  terms.  If  for  words,  abscissa , ordinate , 
vertex , in  Rule  B,  be  substituted  longitude,  latitude,  station , 
that  rule  applies  technically  to  the  case  of  land  surveys. 

Example. — Recently  a survey  of  a tract  of  one  hundred  and  fifty 
square  miles  of  coal  land  in  Tennessee  was  made  on  the  co-ordinate 
system  ; that  is,  every  corner  of  the  old  grants,  of  the  subsequent 
sales  out  of  them,  and  of  the  older  adverse  claims  within  them, 
and  every  station  of  a road  or  other  traverse,  as  well  as  all 
important  points,  as  coal  mines,  outcrops,  bridges,  crossroads, 
springs,  villages,  etc.,  etc.,  was  fixed  in  position  with  respect 
to  an  assumed  point,  by  means  of  rectangular  co-ordinates 
directed  due  north  and  east.  Thus,  any  parcel  of  land  could 
be  easily  plotted,  when  wanted,  in  any  of  the  geological  or 
other  maps ; while  merely  a glance  at  the  table  served  to 
indicate  its  position  in  the  district.  When  the  area  of  such  a 
piece  was  required,  it  was  only  necessary  to  proceed  as  in  the 
following  example: 


Station. 

Total 

latitude. 

Total 

longitude. 

Difference 

between 

alternate 

longitudes. 

Double 

areas. 

A 

7087  ft. 

94851  ft. 

-r  1201 

8511487 

B 

10020 

97403 

— 6518 

— 65310360 

0 

8181 

101369 

— 5752 

—47057112 

I) 

5012 

103155 

-f  2765  ‘ 

13858180 

E 

2873 

98004 

+ 8304 

23857392 

Area  in  sq.  ft. 


GO 140413 


130 


ELEMENTS  OF  SURVEYING. 


[l300K  III. 


If  the  latitudes  were  larger  than  longitudes,  differences  of 
latitudes  should  be  made  the  factors  with  total  longitudes. 

Note. — The  total  latitude  of  a station  with  respect  to  any 
assumed  point,  is  the  distance,  measured  on  a meridian  through  the 
assumed  point,  from  the  point  to  the  foot  of  a perpendicular  drawn 
from  the  station  to  the  meridian  ; and  the  total  longitude  is  the  dis- 
tance of  the  station  from  the  meridian  through  the  assumed  point. 

To  find  the  area  of  a polygon  surveyed,  from  the  field  notes 
of  the  bearings  and  distances  of  its  sides. 

After  finding  the  latitude  and  departure  of  each  course  and 
balancing  the  survey,  as  already  explained,  make  a column  of 
total  latitudes  of  the  various  stations  referred  to  any  one  of  them 
as  origin.  This  selection  of  origin  reduces  one  double  area 
product  to  zero.  Now,  instead  of  calculating  the  total  longitudes 
of  all  stations  and  finding  the  differences  between  alternate  ones, 
according  to  Rule  B in  formula  (7),  we  may  obtain  the  same 
result  in  simpler  manner  by  adding  the  departures  of  each  pair 
of  adjacent  courses.  Modified  in  this  way  the  operation  becomes 
governed  by  the  following  rule : 

Rule  0. — Multiply  the  total  latitude  of  each  station 
by  the  sum  of  the  departures  of  the  two  adjacent  courses. 
The  algebraic  half  sum  of  these  products  is  the  area. 


EXAMPLE. 


Stations. 

Rearing. 

Distance. 

Latitude. 

Departure. 

Total 

Latitude. 

Adjacent 

Departures. 

Double 

Areas. 

N + 

S- 

E + 

W- 

A 

N 85°  00'  E 

2.70 

2.21 

1.55 

R 

N 83  ’ 30'  E 

1.29 

.15 

1.28 

2.21 

2.83 

6.2543 

c 

S 57"  00'  K* 

2.22 

1.21 

1.86 

2.36 

3.14 

7.4101 

I) 

S 34°  15'  W 

3.55 

2.93 

2.00 

1.15 

-0.14 

-0.1610 

E 

N 50  " 80'  W 

3.28 

1.78 

2.09 

-1.78 

-4.09 

8.3482 

2 )_2t  .8519 

Square  chains  . . . 10.9259 


SEC.  V.] 


MAGNETIC  DECLINATION. 


131 


To  find  tho  total  latitude  of  each  station,  add  to  total  latitude 
of  preceding  station  the  latitude  of  preceding  course.  If  the 
latitude  of  last  station,  found  in  this  way,  be  equal  to  the 
latitude  of  last  course  with  reversed  sign,  the  work  is  correct. 
However,  the  latitude  of  first  station,  or  station  taken  as  origin, 
is  always  zero ; of  last  station  it  is  always  the  latitude  of  last 
course  with  reversed  sign ; and  of  second  station  it  is  always  the 
latitude  of  first  course.  To  find  the  adjacent  departures , add  the 
departures  of  the  two  courses,  one  on  each  side  of  the  station. 


SECTION  V. 

MAGNETIC  DECLINATION  OR  VARIATION  OF  THE 

NEEDLE. 

The  following  articles,  149-164,  are  essentially,  and  so  far 
as  .practicable  verbatim,  from  papers  of  the  U.  S.  Coast  and 
Geodetic  Survey. 

149.  The  magnetic  declination  at  any  place  is  the  angle 
which  the  compass-needle,  when  it  is  correctly  constructed  and 
freely  suspended,  makes  with  the  true  meridian.  The  true 
meridian  is  fixed,  but  the  declination  varies  because  the  direc- 
tion in  which  the  needle  points  is  in  a continuous  state  of 
change.  Therefore,  whenever  a measure  of  the  declination  of 
the  needle  is  taken,  the  exact  time  (year,  day  of  month,  and 
hour  of  the  observations)  should  be  recorded  as  well  as  the 
geographical  position  of  the  place,  or  its  latitude  and  longitude 
expressed  to  the  nearest  minutes  of  arc. 

150.  T1  ic  declination  is  called  “West”  when  the  north  end 
of  tho  needle  points  to  the  west  of  the  true  meridian,  and  it  is 


132 


ELEMENTS  OF  SURVEYING. 


[ BOOK  III. 


called  “East”  when  the  north  end  of  the  needle  points  east  of 
the  true  meridian.  In  order  to  give  an  idea  of  the  amount  of 
the  declination  at  present  observable  within  the  limits  of  the 
United  States  we  instance  the  following  places  at  or  near  which 
it  reaches  extreme  value,  which  are  given  to  the  nearest  whole 
degree  (1878)  : 

At  Eastport,  Me.,  declination  18°  west. 

At  the  mouth  of  the  Rio  Grande,  Texas,  8°  east. 

At  San  Diego,  Cal.,  14°  east. 

At  Sitka,  Alaska,  29°  east. 

At  Fort  Yukon,  Alaska,  36°  east. 

151.  The  accuracy  with  which  the  declination  may  be  deter- 
mined depends  chiefly  upon  the  instrumental  means,  but  also, 
and  in  a great  measure,  upon  the  care  taken  in  the  use  of  the 
instruments  and  the  selection  of  the  proper  methods  and  times 
for  observing. 

Omitting  any  detailed  notice  of  the  irregular  variations  to 
which  the  magnetic  needle  is  subject,  it  becomes  important  for 
the  purposes  of  the  surveyor  to  refer  particularly  to  the  changes 
which  have  a special  bearing  upon  his  observations.  These  are 
the  daily  variation  and  the  secular  variation. 

152.  The  Daily  Variation. — It  has  been  found  that  at 
about  the  time  of  sunrise  the  north  end  of  the  needle  has  a slow 
motion  towards  the  east  which  soon  ceases.  The  needle  is  then 
said  to  be  at  its  eastern  elongation ; its  north  end  then  begins  a 
retrograde  motion  towards  the  west,  and  at  about  one  o’clock 
in  the  afternoon  reaches  the  point  at  which  it  is  said  to  be  at 
its  western  elongation,  after  which  it  again  turns  back  towards 
the  east. 

The  times  at  which  the  needle  reaches  its  eastern  and 
western  elongations  vary  with  the  seasons  of  the  year  (with 


SEC.  V.] 


MAGNETIC  DECLINATION. 


133 


the  sun’s  declination),  happening  a little  earlier  in  summer 
than  in  winter. 

The  angular  range  between  the  eastern  and  western  elonga- 
tions varies  also  with  the  season  of  the  year. 

The  average  position  of  the  needle  for  the  day  is  called  the 
mean  magnetic  meridian. 

At  about  six  o’clock  in  the  evening  (and  for  about  an  hour 
before  and  after),  throughout  the  year,  the  position  of  the  needle 
coincides  very  nearly  with  the  mean  magnetic  meridian,  and  this, 
therefore,  is  the  time  most  favorable  for  making  observations  to 
obtain  at  once  the  mean  declination. 

153.  For  reducing  the  direction  of  the  needle  observed  at 
other  hours  to  the  mean  magnetic  meridian  the  following  table  is 
furnished.  It  gives  to  the  nearest  minute  the  variations  of  the 
needle  from  its  average  position  during  the  day,  for  each  hour 
in  the  day  for  the  four  seasons  of  the  year. 

Table  for  reducing  the  observed  declination  to  the  mean  decli- 
nation of  the  day. 


The  needle  points  east  of  the 
mean  magnetic  meridian. 

The  needle  points  west  of  the  mean  magnetic 
meridian. 

A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

A.M. 

Noon. 

P.  M. 

P.M. 

P.M. 

P.  M. 

P.M. 

P.M. 

h. 

h. 

h. 

h. 

h. 

h. 

a 

o 

A. 

h. 

h. 

h. 

h. 

h. 

Hour 

6 

7 

8 

9 

10 

11 

o 

£ 

1 

2 

3 

4 

5 

6 

/ 

, 

, 

, 

r 

! 

, 

, 

/ 

, 

/ 

, 

Spring 

3 

4 

4 

3 

1 

1 

4 

5 

5 

4 

3 

2 

1 

Summer.. .. 

4 

5 

5 

4 

1 

2 

4 

6 

5 

4 

3 

2 

1 

Autumn  . . . 

2 

3 

3 

2 

0 

2 

3 

4 

3 

2 

1 

1 

0 

Winter 

1 

1 

2 

2 

1 

0 

2 

3 

3 

2 

1 

1 

1 

0 

154.  It  appears  from  observations  of  the  daily  fluctuation  of 
the  declination  that  the  mean  of  the  extreme  easterly  and 


134 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


westerly  positions  m any  one  day  approaches  nearly  (within  half 
a minute)  to  the  mean  position  of  the  day,  as  derived  from 
hourly  observations  continued  day  and  night.  Since  corrections 
to  observed  declinations  to  refer  them  to  the  mean  of  the  day  are 
generally  very  unsatisfactory,  it  is  recommended  to  observe  the 
declination  for  any  one  day  at  the  epochs  of  the  eastern  magnetic 
elongation  and  of  the  western  magnetic  elongation  and  to  take 
the  mean  position  as  representing  the  declination  for  that  day. 
The  epochs  of  extreme  positions,  as  observed  at  Philadelphia, 
Washington,  and  Key  West,  apply,  with  comparatively  small 
changes,  to  nearly  all  places  within  the  United  States  and  may 
be  stated  to  he  as  follows:  Referring  to  the  north  end  of  the 
magnet,  the  morning  eastern  elongation  occurs,  on  the  average, 
from  May  to  September,  inclusive,  about  7-J  a.m.  ; in  March, 
April,  and  October,  about  8 a.m.;  in  November,  about  8J  a.m., 
and  in  December,  January,  and  February,  about  9 a.m.;  earliest 
time  in  August,  about  a.m.;  latest  in  January,  about  9 a.m. 
These  epochs,  however,  are  subject  to  great  fluctuations  and 
cannot  be  depended  upon  in  any  one  case  within  one  hour  and 
frequently  they  cannot  be  recognized  at  all,  either  on  account  of 
the  small  range  of  the  daily  fluctuation — the  amount  of  which  in 
winter  is  but  one-half,  nearly,  of  the  amount  in  summer — which 
is  easily  disguised  by  small  irregularities,  or  on  account  of 
disturbances,  which  reach  their  maxima  in  September  and 
October,  and  generally  are  more  predominant  in  winter  than  in 
summer.  The  afternoon  ivestern  elongation  occurs,  on  the 
average,  about  1}  p.m.  from  May  to  November,  inclusive,  and 
about  If  p.m.  in  the  remaining  months;  also,  earliest  in  Septem- 
ber— some  minutes  before  1 p.m. — and  latest  in  January,  about 
If  p.m.  The  afternoon  epoch  is  subject  to  less  fluctuation  than 
the  morning  epoch. 

155  The  Secular  Variation  of  the  magnetic  declination 


SEC.  V.] 


MAGNETIC  DECLINATION. 


135 


is  a subject  of  the  greatest  importance  to  surveyors.  It  mani- 
fests itself  by  a gradual  change  in  one  direction,  which  at  first 
increases  slowly,  then  more  rapidly,  diminishing  again  after- 
ward until  the  needle  becomes  stationary  and  subsequently 
returns  by  similar  changes  to  its  former  position,  the  whole 
period  extending  over  nearly  two  and  a half  centuries.  Thus, 
at  Philadelphia  the  declination  was  8f°  west  in  1700,  whence  it 
diminished  until  in  1800  it  reached  a minimum  2°.l  (2°  6'), 
and  increased  again  to  6°. 8 in  1880.  At  present  all  along 
the  Atlantic  and  Gulf  coasts  the  effect  of  the  secular  variation 
is  to  increase  west  declinations  or  to  decrease  east  declinations 
by  from  2'  to  5',  but  on  the  Pacific  coast  the  effect  is  opposite 
in  direction,  viz.,  increasing  east  declinations  by  from  1'  to  3'. 
In  Alaska,  however,  there  are  indications  of  a decrease  of  east 
declination. 

156.  Lying  between  the  region  in  which  the  variation  of 
the  needle  is  west  and  that  in  which  such  variation  is  east, 
is  a line  of  no  variation , on  which  the  magnetic  meridian 
coincides  with  the  true  meridian. 

The  north  (or  seeking)  end  of  the  needle  always  inclines 
toward  the  line  of  no  variation : hence,  for  all  points  east  of 
this  line,  the  variation  is  West;  and  for  all  points  west  of  it, 
the  variation  is  East. 

This  line  is  not  a fixed  line,  but  changes  its  position  from 
year  to  year.  By  referring  to  a map,  such  line  for  the  United 
States  may  be  traced.  In  1870,  it  crossed  Sault  St.  Marie  at  the 
lower  end  of  Lake  Superior,  passed,  very  nearly,  through  Cleve- 
land in  the  State  of  Ohio,  Raleigh  in  North  Carolina,  and 
passed  into  the  Atlantic  Ocean  near  Wilmington,  North  Caro- 
lina. In  1875,  it  crossed  Lake  Superior  and  entered  Michigan 
at  White  Pish  Point,  passed  thence,  very  nearly,  through  Bay 
City  Michigan,  Oberliu  Ohio,  Parkersburgh  West  Virginia, 


136 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


thence,  with  a slight  curve  to  the  southwest,  to  Fayetteville 
North  Carolina,  and  thence  into  the  Atlantic  Ocean  a little 
to  the  west  of  Cape  Fear.  The  line  of  no  variation  is  now 
moving  quite  rapidly  to  the  south  and  west,  and  therefore  it 
is  that  ivest  declinations  of  the  needle  are  increasing,  and  east 
declinations  decreasing. 

157.  The  U.  S.  Coast  and  Geodetic  Survey  has  made  a 
collection  of  magnetic  declinations  observed  at  various  places  in 
the  United  States  and  elsewhere.  Mr.  Charles  A.  Schott,  chief 
of  the  Computing  Division  of  the  Survey,  has  tabulated  the 
observed  declinations,  and  deduced  from  them  a Table  of  For- 
mulae. This  table  is  here  given  so  far  as  it  relates  to  localities 
in  the  United  States.  The  localities  are  arranged  geographically, 
as  far  as  practicable,  and  their  positions  are  given  by  latitude  and 
longitude  (west  of  Greenwich). 


Formulae  expressing  the  magnetic  declination  at  various  places 
in  the  United  States. 


Locality. 

Latitude. 

Longitude. 

Expression  for  Magnetic  Declination. 

o / 

O / 

o o o 

Portland,  Me 

43  38.8 

70  16.6 

D—  +10.72  + 2.68  sin  (1 .33  m + 24.1). 

Burlington,  Vt 

44  28.2 

73  12.3 

J)—  +10.81  + 3.65  sin  (1.30  m—  20.5) 

+ 0.18  sin  (7.0  m + 132). 

Rutland  Vt 

43  36.5 

43  04.8 

72  55.5 

D— f 10.03  + 3.82  sin  (1.5  m—  24.3). 

Portsmouth,  N.  II 

70  43.0 

D—  +10.63  + 3.17  sin  (1.44  m — 4.7). 

Newburyport,  Mass.. 

42  48.4 

70  49.0 

D=  + 10.07  + 3.10  sin  (1.4  m + 1.9). 

Salem,  Mass 

42  31.9 

70  52.5 

D=+  9.80  + 3.61  sin  (1.50  m—  1.0). 

Boston,  Mass — 

42  21.5 

71  03.8 

D=  + 9.52  + 2.93  sin  (1.30  m + 5.0). 

Cambridge  Mass 

42  22.9 

71  07.7 

D=+  9.58  + 2.69  sin  (1.3  m + 7.0) 

+ 0.18  sin  (3.2  m + 44). 

Nantucket,  Mass. . . . 

41  17.0 

70  06.0 

D=+  9.29  + 2.78  sin  (1.35  m + 5.5). 

Providence,  R.  I 

41  49.5 

71  24.1 

D=  + 9.10  + 2.99  sin  (1.45  m—  3.4) 

+ 0.19  sin  (7.2  m + 116). 

Hartford,  Conn 

41  45.9 

72  40.4 

1)  + 8.06  + 2.90  sin  (1.25  m— 26.4). 

New  Haven,  Conn  . . . 

41  18.5 

72  56.7 

D=  + 7.78  + 3.11  sin  (1.40  m—  22.1). 

SEC.  V.] 


MAGNETIC  DECLINATION. 


137 


Locality. 

Latitude. 

Longitude. 

Expression  for  Magneto  declination. 

Albany,  N.  Y 

42  39.2 

73  45.8 

D=  + 8.17+3.02  sin  (1.44  m—  8.3). 

Oxford,  N.  Y 

42  26.5 

75  40.5 

D=  + 6.19  + 3.24  sin  (1.35  m — 18.9). 

Buffalo  N Y 

42  52.8 

78  53.5 

D=+  3.66  + 3.47  sin  (1.4  m — 27.8). 

Erie,  Pa. 

42  07.8 

80  05.4 

D=  + 2.26  + 2.71  sin  (1  55  m—  29.7). 

Cleveland,  Ohio 

41  30.3 

81  42.0 

D=+  0.10  + 2 07  sin  (1.40  m — 6.2). 

Detroit,  Mich  

42  20.0 

83  03.0 

D= — 0.97  + 2.21  sin  (1.50  m—  15.3). 

Saint  Louis,  Mo 

38  38.0 

90  12.2 

D=— 7.15  + 2.33  sin  (1.4  m— 20.1).* 

New  York,  N.  Y 

40  42.7 

74  00.0 

D=+  6.40  + 2.29  sin  (1.6  m—  5.5) 

+ 0.14  sin  (6.3  m + 64). 

Hatborough,  Pa 

40  12 

75  07 

D=  + 5.23  + 3.28  sin  (1.54  m—  13.2) 

+ 0.22  sin  (4.1  m + 157). 

Philadelphia,  Pa 

39  56.9 

75  09.0 

D=+  5.38  + 3.29  sin  (1.55  m— 23.9) 

+ 0.39  sin  (4.0  m + 161). 

Harrisburg,  Pa 

40  15.9 

76  52.9 

D=  + 2.93+2.98  sin  (1.50  m+  0.2). 

Baltimore,  Md 

39  17.8 

76  37.0 

D=+  3.20  + 2.57  sin  (1.45m—  21.2). 

Washington,  D.  C 

38  53.3 

77  00.6 

D—  + 2.47  + 2.47  sin  (1.40  m-  14.6). 

Cape  Henry,  Ya 

36  55.5 

76  00.5 

D=+  2.54  + 2.41  sin  (1.50  m— 35.4). 

Charleston,  S.  C 

32  46.6 

79  55.8 

D=-  2.14+2.74  sin  (1.35m  1.3). 

Savannah,  Ga 

Key  West,  Fla 

32  04.9 

24  33.5 

81  05.5 

81  48.5 

D=-  2.54  + 2.32  sin  (1.5  m-  28.6). 

D = — 3.90  + 2.93  sin  (1.4  m—  33.5). 

Mobile,  Ala 

30  41.4 

88  02.5 

D= — 4.40  + 2.69  sin  (1.45  m—  76.4). 

New  Orleans,  La 

29  57.2 

90  03.9 

D= — 5.61  + 2.57  sin  (1.4  m-  61.9). 

San  Diego,  Cal 

32  42.1 

117  14.3 

D= — 12.54  + 1.64  sin  (1.2  m— 180.0). 

Monterey,  Cal 

36  36.1 

121  53.6 

D= — 12.82  + 3.54  sin  (1.0  m-142.9). 

San  Francisco,  Cal. . . 

Cape  Disappointment, 

37  47.5 

122  27.2 

D=— 13.34  + 3.23  sin  (1.00  m-130.3). 

Wash  TV.r 

46  16.7 

124  02.0 

D=— 20.72  + 2.81  sin  (1.2  m— 188.8). 

Sitka,  Alaska 

57  02.9 

135  19.7 

D=— 26.72  + 2.41  sin  (1.6  m-107.1). 

Unalashka,  Alaska. .. 

53  52.6 

166  31.5 

D= — 18.34  + 1.45  sin  (1.4  m— 67.8). 

158.  The  epoch  to  which  the  formulae  refer  is  1850  ; hence, 
in  the  “ expression  for  magnetic  declination,” 
m — t — 1850. 

To  illustrate  the  use  of  the  table.  Let  it  be  required  to  find  the 
declination  of  the  needle  at  Albany,  N.  Y.,  in  August,  1879, 
or  1879.6. 


Approximate  expression. 


138 


ELEMENTS  OF  SURVEYING. 


[BOOK  III 


The  tabular  expression  for  magnetic  declination  at  Albany  is 
D = + 8°.17  + 3°.02  sin  (1.44°  m-8°.3). 
m = 1879.6  — 1850  = 29.G. 

1.44°  m = 42°.G24. 

...  1.44°  s°.3  = 34°.324. 

Natural  sin  (34°.324)  = .5638  ; 

hence,  D = -f8o.17  + 3°.02  x .5G38  = + 9°.87; 

which  gives  the  computed  declination  of  the  needle  at  Albany  in 
1879. 6, .as  9°.87  (west,  since  the  result  is  plus),  which  differs  from 
the  observed  declination  at  that  time  but  one-hundredth  of  a 
degree. 

If  the  time  for  which  the  declination  is  desired  is  prior  to 
1850,  then  m will  be  negative.  For  example,  let  the  declination 
at  Albany  for  1836.8  be  required  ; then 

m = 1836.8—1850  = —13.2, 
and  1°.44  m = 1°.44  x (-13.2)  = -19°.008, 

and  1°.44  m — 8°.3  = — 27°.308, 

and  sin  (— 27°.308)  = —.45877; 

D = 8°.17  + 3°.02  sin  (-27°.308)  = 8°.17  — 1°.39  = +6°. 78; 

which  agrees  exactly  with  the  observed  declination. 

The  student  is  referred  to  Mr.  Schott’s  paper,  Appendix 
No.  9,  U.  S.  Coast  and  Geodetic  Survey  Report  for  1879,  for 
valuable  tables  of  Magnetic  Declinations. 

159.  From  the  same  paper  is  taken  the  following  table  of 
computed  annual  changes  in  the  declination  of  the  magnetic 
needle  for  1870,  1880,  and  1885,  expressed  in  minutes  of  arc,  a 
-f  sign  indicating  north  end  of  needle  moving  westward,  a — 
sign  indicating  north  end  moving  eastward: 


SEC.  V.j 


MAGNETIC  DECLINATION. 


130 


TABLE. 


Locality. 

Annual  change. 

In  1870. 

In  1880. 

In  1885. 

Portland,  Me. 

/ 

+ 2.4 

+ 1.6 

+ 1.2 

Burlington,  Vt 

+ 5.0 

+ 6.0 

+ 5.8 

Rutland,  Vt 

+ 6.0 

+ 5.6 

+ 5.3 

Portsmouth,  N.  II 

+ 4.4 

+ 3.7 

+ 3.3 

Newburyport,  Mass 

+ 3.9 

+ 3.3 

+ 2.9 

Salem,  Mass 

+ 5.0 

+ 4.1 

+ 3.5 

Boston,  Mass 

+ 3.4 

+ 2.9 

+ 2.5 

Cambridge,  Mass 

+ 2.9 

+ 2.1 

+ 1.8 

Nantucket,  Mass 

+ 3.3 

+ 2.7 

+ 2.4 

Providence,  R.  I 

+ 3.8 

Hartford,  Conn 

+ 3.8 

+ 3.7 

+ 3.6 

New  Haven,  Conn 

+ 4.6 

+ 4.3 

+ 4.1 

Albany,  N.  Y 

+ 4.3 

+ 3.7 

+ 3.4 

Oxford,  N.  Y 

+ 4.5 

+ 4.3 

+ 4.0 

Buffalo,  N.  Y 

+ 5.1 

+ 5.0 

+ 4.8 

Erie,  Pa 

+ 4.4 

+ 4.2 

+ 4.0 

Cleveland,  Ohio 

+ 2.8 

+ 2.5 

+ 2.2 

Detroit,  Mich 

+ 3.4 

+ 3.0 

+ 2.8 

Saint  Louis,  Mo.  

+ 3.4 

+ 3.2 

+ 3.0 

New  York,  N.  Y 

+ 2.4 

+ 2.5 

+ 2.6 

Hatborough,  Pa. 

+ 4.6 

+ 4.5 

Philadelphia,  Pa 

+ 4.9 

+ 4.9 

+ 5.3 

Harrisburg,  Pa 

+ 4.1 

+ 3.3 

+ 2.8 

Baltimore,  Md 

+ 3.9 

+ 3.6 

+ 3.2 

Washington,  D.  C. 

+ 3.5 

+ 3.2 

+ 3.0 

Cape  Henry,  Va 

+ 3.8 

+ 3.7 

+ 3.6 

Charleston,  S.  C 

+ 3.5 

+ 3.0 

+ 2.7 

140 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


Locality. 

Annual  change. 

In  1870. 

In  1880. 

In  1885. 

Savannah,  Ga 

+ 3.G 

+ 3.5 

+ 3.3 

Key  West,  Fla 

+ 4.3 

+ 4.2 

+ 4.1 

Mobile,  Ala 

+ 2.8 

+ 3.4 

+ 3.7 

New  Orleans,  La 

+ 3.1 

+ 3.5 

+ 3.7 

San  Diego,  Oal 

— 1.9 

-1.7 

— 1.6 

Monterey,  Cal 

— 2.0 

— 1.5 

—1.1 

San  Francisco,  Cal 

— 1.0 

— 0.5 

—0.3 

Cape  Disappointment,  W.  Ter.. 

—3.4 

-3.1 

-2.7 

Sitka,  Alaska .... 

+ 1.0 

+ 2.1 

+ 2.5 

Unalashka,  Alaska 

+ 1.6 

+ 1.9 

+ 2.0 

160.  It  will  be  observed  that  the  amount  of  change  is  by  no 
means  the  same,  even  in  places  not  far  remote  from  each  other, 
as  New  York  and  Philadelphia. 

In  grouping  together  a table  of  the  present  rate  of  change, 
much  allowance  must  therefore  be  made  for  possible  local  pecu- 
liarities that  have  not  been  ascertained. 

A surveyor  should,  at  the  time  and  place  of  survey,  deter- 
mine the  true  meridian  and  thence  the  magnetic  declination  for 
record  with  his  survey. 

Method  of  ascertaining  the  Declination. 

161.  The  best  practical  method  of  determining  the  true 
meridian  of  a place,  is  by  observing  the  north  star,  Polaris. 
If  this  star  were  precisely  at  the  point  in  which  the  axis  of  the 
earth  prolonged  pierces  the  heavens,  then  the  intersection  of 
the  vertical  plane  passing  through  it  and  the  place,  with  the 
surface  of  the  earth,  would  be  the  true  meridian.  But  the 


SEC.  V.] 


MAGNETIC  DECLINATION. 


141 


star  being  at  a distance  from  the  pole  equal  to  1°  30'  nearly, 
it  performs  a revolution  about  the  pole  in  a circle,  the  polar 
distance  of  which  is  1°  30'  nearly,  and  the  time  of  revolution  is 
23  hours  and  5G  minutes. 

To  the  eye  of  an  observer  this  star  is  continually  in  motion, 
and  is  due  north  but  twice  in  23  hours  and  56  minutes,  and  is 
then  said  to  be  on  the  meridian.  When  it  departs  from  the 
meridian,  it  apparently  moves  east  or  west  for  5 hours  and 
59  minutes,  and  then  returns  to  the  meridian  again.  When  at 
its  greatest  distance  from  the  meridian,  east  or  west,  it  is  said  to 
be  at  its  greatest  eastern  or  western  elongation. 

162.  The  western  elongations  from  the  beginning  of  April  to 
the  end  of  September,  and  the  eastern  from  the  beginning  of 
October  to  the  end  of  March  occur  in  the  day-time.  If  it  be 
necessary  to  determine  the  meridian  at  that  particular  season  of 
the  year,  and  at  night,  let  5 hours  and  59  minutes  be  added  to  or 
subtracted  from  the  time  of  greatest  eastern  or  western  elongation, 
and  the  observation  be  made  at  night  when  the  star  is  on  the 
meridian. 

163.  The  angle  which  the  meridian  plane  makes  with  the 
vertical  plane  passing  through  the  pole-star  when  at  its  greatest 
eastern  or  western  elongation,  is  called  the  Azimuth  of  Polaris. 

The  following  extract,  Art.  164,  with  the  tables,  is  from  the 
Report  of  the  U.  S.  Coast  and  Geodetic  Survey  for  1881 : 

164.  The  following  tables  of  the  times  and  azimuths  of  Po- 
laris  when  at  elongation  have  been  prepared  for  the  benefit  of  those 
surveyors  and  others  who  may  prefer  to  make  use  of  the  pole-star 
for  their  determination  of  the  true  meridian,  and  whose  instru- 
mental outfit  for  the  measure  of  the  declination  may  be  limited  to 
a compass  with  sights  or  to  a small  theodolite  with  compass-needlo 
attached,  and  who  may  be  without  a chronometer. 


142 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


The  method  was  recommended  to  surveyors  by  Dr.  Charles 
Davies  in  the  revised  edition  of  his  work  on  surveying,  and  a 
description  of  it  still  forms  part  of  the  instructions  of  the  Com- 
missioner of  the  General  Land  Office  to  the  surveyors-general  of 
public  land  of  the  United  States  (editions  of  1855,  1871,  and  1878). 
The  tables  given  in  these  instructions  have  either  become  obsolete 
from  lapse  of  time  or  are  not  sufficiently  extended  for  future  use. 
They  were,  therefore,  recomputed,  and  in  their  present  form  and 
with  the  rules  given  for  interpolation  will  be  found  to  possess 
greater  accuracy  than  any  similar  tables  previously  published. 
The  tables  include  all  elongations  whether  occurring  by  day  or 
night.  Polaris  may  be  observed  in  day-time  when  the  sun  is  not 
too  high  even  with  moderately  powerful  telescopes ; besides,  a 
complete  table  facilitates  interpolation. 

Mean  local  time  (astronomical,  counting  from  noon)  of  the 
elongations  of  Polaris. 

[The  table  answers  directly  for  the  year  1885,  for  latitude  + 40°  and  for  longitude  6 hours 
west  of  Greenwich.] 


Date. 

Eastern  elongation. 

Western  elongation. 

Jan. 

1 

h. 

0 

m. 

35.3 

h. 

12 

m. 

24.6 

it 

15 

23 

3G.1 

11 

29.3 

Feb. 

1 

22 

29.0 

10 

22.2 

tt 

15 

21 

33.7 

9 

27.0 

Mar. 

1 

20 

38.5 

8 

31.8 

(t 

15 

19 

43.4 

7 

36.6 

Apr. 

1 

18 

36.4 

6 

29.7 

tt 

15 

17 

41.4 

5 

34.7 

May 

1 

16 

38.6 

4 

31.8 

it 

15 

15 

43-7 

3 

36.9 

June 

1 

14 

37.1 

2 

30.3 

it 

15 

13 

42.2 

1 

35.4 

SEC.  V.] 


MAGNETIC  DECLINATION. 


143 


Date. 

Eastern  elongation.  Western  elongation. 

July 

h. 

m. 

h. 

m. 

1 

12 

39 

6 

0 

32.8 

(( 

15 

11 

44 

.7 

23 

34.0 

Aug. 

1 

10 

38 

.2 

22 

27.5 

ii 

15 

9 

43 

.3 

21 

32.6 

Sept. 

1 

8 

30 

.7 

20 

26.0 

a 

15 

7 

41 

.7 

19 

31.1 

Oct. 

1 

6 

38. 

.9 

18 

28.2 

15 

5 

43. 

.9 

17 

33.2 

Nov. 

1 

4 

?> 

CO 

.0 

16 

26.4 

a 

15 

3 

41. 

9 

15 

31.3 

Dec. 

1 

2 

00 

CO 

9 

14 

28.2 

(£ 

15 

1 

43. 

6 

13 

33.0 

N.  B. — To  refer  the  tabular  times  to  any  year  (limit  about 
10  years)  subsequent  to  the  epoch,  add  0m.35  for  every  year.  For 
years  previous  to  epoch  subtract  0m.35  for  every  year. 

To  refer  the  tabular  times  to  any  other  latitude  (between  the 
limits  25°  and  50°),  add  0m.14  for  every  degree  south  of  40°  ; 
subtract  0m.18  for  every  degree  north  of  40°. 

To  refer  the  tabular  times  to  any  year  in  a quadriennium, 

For  first  year  after  a leap  year  the  table  is  perfect. 

For  second  year  after  a leap  year  add  . . . lm.  0 

For  third  year  after  a leap  year  add  . . . 2m.O 

For  a leap  year  and  before  March  1 add  . . 3m.O 

And  for  remainder  of  the  year  subtract  . . lm.O 

For  any  other  than  the  tabular  day  subtract  from  the  tabular 
time  of  elongation  3m.94  for  every  day  elapsed. 

It  will  be  noticed  that  there  occur  two  eastern  elongations  on 
January  9,  and  two  western  elongations  on  July  9. 


Azimuth  (from  the  north)  of  Polaris,  when  at  elongation,  between  the  years  1882  and  1895,  for  different 

latitudes  between  +25°  and  +50°. 


144 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


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+ + 

SEC.  V.]  . MAGNETIC  DECLINATION.  ' 145 

To  find  the  true  Meridian  with  the  Compass. 

165.  1.  Drive  two  posfe  firmly  into  the  ground,  in  a line 

nearly  east  and  west  ; the  uppermost  ends,  after  the  posts  are 
driven,  being  about  three  feet  above  the  surface  and  at  the  same 
level,  and  the  posts  about  four  feet  apart ; then  lay  a plank,  or 
piece  of  timber  three  or  four  inches  in  width,  and  smooth  on 
the  upper  side,  upon  the  posts,  and  let  it  be  pinned  or  nailed, 
to  hold  it  .firmly. 

2.  Prepare  a piece  of  board  four  or  five  inches  square,  and 
smooth  oh  the  under  side.  Let  one  of  the  compass-sights,  or  a 
piece  of  tin  with  a very  small  perforation,  he  fastened  at  right 
angles  to  the  upper  surface  of  the  board,  and  place  a brick  or 
other  weight  upon  the  sight-board  to  keep  it  steady. 

.3.  At  about  twelve  feet  from  the  stakes,  and  in  the  direction 
of  the  pole-star,  let  a plumb  be  suspended  from  the  top  of  an 
■ inclined  stake  or  pole.  The  top  of  the  ‘pole  should  be  of  such 
'a  height  that  the  pole-star  will  appear  about  six  inches  below  it ; 
and  the  plumb  should  be  swung  in  a vessel  of  water  to  prevent 
it  from  vibrating. 

This  being  done,  about  twenty  minutes  before  the  time  of 
elongation,  place  the  board,  to  which  the  compass-sight  is  fastened, 
on  the  horizontal  plank,  and  slide  it  east  or  west,  until  the 
aperture  of  • the  compass-sight,  the  plumb-line,  and  the  star,  are 
brought  into  the  same  range.  Then  if  the  star  depart  from 
the  plumb-line,  move  the  compass-sight,  east  or  west,  along 
•the  timber,  as  the  case  may  be,  until  the  star  shall  attain  its 
greatest  elongation,  when  it  will  continue  behind  the  plumb-line 
for  several  minutes;  and  will  then  recede  from  it  in  the  direc- 
tion contrary  to  its  motion  before  it  became  stationary.  Dur- 
ing this  observation  it  will  be  necessary  to  have  the  plumb- 
line  lighted  ; this  may  be  done  by  an  assistant  holding  a candle 
near  it. 


140 


ELEMENTS  OE  SURVEYING. 


[LOOK  III. 


Now  let  a point  be  fixed  by  a second  plumb-line  placed  at  a 
distance  of  thirty  or  forty  yards  from  the  first,  and  in  the  same 
direction  with  it  and  the  compass-sight.  Mark  the  points 
determined  by  the  two  plumbs,  for  future  reference,  by  driving 
pegs  deeply  into  the  ground  and  marking  the  exact  point  on  each 
by  a small  nail.  The  line  so  determined,  makes,  with  the  true 
meridian,  an  angle  equal  to  the  azimuth  of  the  pole-star;  and, 
from  this  line,  the  variation  of  the  needle  is  readily  determined, 
even  without  tracing  the  true  meridian  on  the  ground. 

Place  the  compass  upon  this  line,  turn  the  sights  in  the 
direction  of  it,  and  note  the  angle  shown  by  the  needle.  Now, 
if  the  elongation  at  the  time  of  observation  was  west,  and  the 
north  end  of  the  needle  is  on  the  west  side  of  the  line,  the 
azimuth  plus  the  angle  shown  by  the  needle,  is  the  true  varia- 
tion. But  should  the  north  end  of  the  needle  be  found  on  the 
east  side  of  the  line,  the  elongation  being  west,  the  difference 
between  the  azimuth  and  the  angle  would  show  the  variation ; 
and  the  reverse  when  the  elongation  is  east. 

It  may  be  stated  that  for  magnetic  purposes  a moderate 
degree  of  accuracy  suffices  in  the  determination  of  the  true 
meridian,  and  a correct  knowledge  of  it  within  1'  will  in  general 
fully  suffice.  It  is  difficult,  even  in  our  middle  latitudes,  to 
determine  the  magnetic  meridian  within  the  limit  of  1'  on 
account  of  the  continuous  fluctuations,  hence  any  greater  accu- 
racy than  this  in  the  astronomic  meridian  would  be  useless. 

166.  A very  near  approximation  to  a true  meridian,  and 
consequently  to  the  variation,  may  be  had,  by  remembering 
that  the  pole-star  very  nearly  reaches  the  true  meridian,  when 
it  is  in  the  same  vertical  plane  with  the  star  Delta  (S)  in  the 
constellation  Cassiopeia. 

Using  the  same  apparatus  as  in  Art.  165,  place  the  “sight- 
hoard”  in  line  with  the  plumb-line  and  the  pole-star,  and 


SEC.  V.] 


MAGNETIC  DECLINATION. 


147 


Jivdg 


IVdjif) 


move  it  to  the  east  as  the  pole-star  moves  ivest , till  Polaris  and 
Delta  both  appear  upon  the 
plumb-line  together ; the  line 
through  the  point  of  sight  and 
the  plumb-line  will  be,  very 
nearly  and  with  sufficient  accu- 
racy, the  true  meridian.  This 
method  is  practicable  only  when  . 

the  star  Delta  is  below  the 

s/ 

pole-star  during  the  night ; • 

when  it  passes  the  meridian 
above  the  pole,  it  is  too  near 
the  zenith  to  be  of  service,  in 
which  case  the  star  Zeta  (£), 
the  last  star  but  one  in  the 
tail  of  the  Great  Bear,  may  be 
used  instead. 

Delta  Cassiopeiae  is  on  the 
meridian  below  the  pole-star 
at  midnight  about  April  10,  and 
is,  therefore,  the  proper  star 
to  use  at  that  date  and  for 
some  two  months  before  and 


Polaris 


IT.Tolt 


after. 

Six  months  later,  the  star 
Zeta  in  the  tail  of  the  Great 
Bear  will  supply  its  place,  and 
is  to  be  used  in  precisely  the 
same  manner. 

Fig.  82  gives  a representa- 
tion, drawn  to  scale,  of  N.  Pole, 
Polaris,  and  the  Constellations 
Cassiopeia  and  Great  Bear;  and 


£• 


s 


AT* 

/•  »/3 

•V 

Q* 

■6* 


Gassi 


(ypcia 


Fig.  82. 


148 


ELEMENTS  OF  SURVEYING. 


[BOOK  III. 


the  line  drawn  through  the  star  Delta  (3)  of  Cassiopeia  and  Zeta 
(£)  of  the  Great  Bear,  represents  those  stars  on  the  meridian  with 
the  pole-star. 

The  method  given  in  this  article  for  finding  the  true  meridian 
cannot  be  used  with  advantage,  on  account  of  the  haziness  of 
the  atmosphere  near  the  horizon,  at  places  below  about  38° 
north  latitude. 

167.  The  variation  of  the  needle  should  always  be  noted  on 
every  survey  made  with  the  compass,  and  then  if  the  land  be  sur- 
veyed at  a future  time,  the  old  lines  can  always  be  re-run. 

In  re-running  the  lines  of  an  old  survey,  a vernier  is  used  for 
setting  off  the  variation  of  the  needle. 

168.  A Vernier  is  a contrivance  for  measuring  smaller  divi- 
sions of  a unit  than  those  into  which  the  line,  to  which  it  is 
applied,  is  divided.  It  is  a graduated  scale  so  arranged  as  to 
cover  an  exact  number  of  equal  spaces  on  the  primary  scale,  or 
limb.  It  is  divided  into  a number  of  equal  parts  greater  by  one 
than  the  number  of  equal  spaces  which  it  covers  on  the  limb. 

The  vernier  may  be  applied  to  any  limb  or  scale  of  equal  parts. 
The  modes  of  its  application  are  extremely  various  ; the  principle, 
however,  is  the  same  in  all,  and  may  be  illustrated  by  a simple 
diagram  (Fig.  83). 


<S  D 10  11  12  13  14  13  16  17  18  19 


c 

1111 ] 1 1 1 1 

n 

0/23436789  /0 

Fig.  83. 


Let  AB  be  any  limb  or  scale  of  equal  parts,  one  of  which  let 
us  suppose  equal  to  ^ of  an  inch.  Let  CD  be  a vernier , equal 
in  length?  say  to  nine  of  these  parts,  and  itself  divided  into  ten 
equal  spaces,  each  one  of  which  is  then  equal  to  nine-tenths  of 
or  , JJ0  of  an  inch.  The  difference  between  a space  on  the 
limb  and  a space  on  the  vernier,  is  therefore  equal  to  one-tenth  of 


SEC.  V.] 


MAGNETIC  DECLINATION. 


149 


-fo,  or  to~o  an  inch.  This  is  the  least  space  that  can  be 
measured  by  means  of  the  vernier,  and  is  called  the  least  count ; 
hence. 

The  least  count  of  a vernier  is  equal  to  one  of  the  equal 
divisions  of  the  limb  divided  by  the  number  of  spaces  on  the 
vernier. 

169.  The  true  reading  of  an  instrument,  for  any  position  of 
the  vernier,  expresses  the  distance  from  the  point  where  the 
graduation  on  the  limb  begins,  marked  0,  to  the  0 point  of 
the  vernier.  In  the  diagram,  that  distance  is  expressed  by 
nine  units  of  the  limb,  or  9. 

If,  now,  the  vernier  be  moved  till  the  division  1 coincides 
with  the  division  10  of  the  limb,  the  0 point  will  have  advanced 
along  the  limb  a distance  equal  to  ^ of  one  space,  and,  if  we  call 
each  space  the  reading  will  become  9 + ^#.  If  we  again  move 
the  vernier  till  the  division  2 coincides  with  the  division  11  of 
the  limb,  the  0 point  will  have  advanced  an  additional  distance, 
equal  to  -fab,  and  the  reading  becomes  9 + ^#  ; when  3 coincides 
with  division  12,  the  reading  will  become  9-f-^-£,  and  so  on,  till 
finally,  when  the  point  10  coincides  with  19  of  the  limb,  the 
distance  9 will  have  been  increased  by  \§b,  and  will  become  10, 
as  it  should,  since,  in  that  case,  the  0 point  will  have  been 
moved  a whole  space,  and  will  coincide  with  the  division  10 
of  the  limb.  Hence,  the  following  rule  for  reading  an  instru- 
ment which  has  a vernier  : 

Read  the  scale,  in  the  direction  of  the  graduation,  up  to  the 
line  preceding  the  0 of  the  vernier  ; this  gives  the  number  of 
whole  units  of  the  scale.  Look  along  the  vernier  and  find  which 
of  its  lines  coincides  with  a line  of  the  scale;  this  line  of  the 
vernier  gives  the  number  of  fractional  parts  of  one  unit  of  the 
scale  to  be  added  to  the  former  reading. 


150 


ELEMENTS  OF  SURVEYING. 


[ROOK  III. 


170.  In  order  to  read  a vernier  correctly,  especially  if  it  be 
one  with  which  we  are  not  familiar,  it  is  necessary  to  estimate, 
by  the  eye  alone , the  fractional  part  of  one  unit  to  be  added  ; 
then  if  the  vernier  reading  is  nearly  the  same  as  the  estimate , 
we  may  record  the  reading  with  confidence ; but  if  the  estimate 
and  the  vernier  reading  disagree  largely,  then  the  cause  of  such 
disagreement  will  probably  be  a false  reading  of  the  vernier. 


171.  In  a vernier  compass,  the  vernier  is  attached,  as 
shown  in  the  figure.  A small  arc  HI  is  described  on  the  bar 
AB,  having  its  centre  at  the  centre  of  the  compass-box.  This 
arc  is  divided  to  degrees,  and  sometimes  to  the  parts  of  a degree. 
The  vernier  is  permanently  attached  to  the  compass-box. 

When  the  0 point  of  this  vernier  coincides  with  the  0 point 
of  the  graduated  arc  HI,  the  north  and  south  line  of  the  com- 
pass-box lies  in  the  plane  of  the  sights. 

The  compass-box  is  turned  about  its  centre,  without  moving 


SEC.  V.] 


MAGNETIC  DECLINATION. 


151 


the  plate  AB,  by  means  of  the  milled  screw  L : and  is  fastened 
to  the  plate  AB,  by  a clamping  nut  underneath  the  main 
plate. 

172.  To  set  off  the  variation  of  the  needle  on  a vernier 
compass,  turn  the  north  end  of  the  compass  plate  by  the  tangent 
screw  L of  the  vernier  (see  Fig.  84),  over  the  number  of  degrees 
in  the  variation,  to  the  left  (the  observer  is  supposed  to  be 
standing  at  the  south  end  of  the  plate  and  looking  towards 
the  north  end),  if  the  variation  is  ivesterly,  to  the  right  if 
it  is  easterly. 

173.  To  run  out  lines  of  old  deeds,  set  the  compass  upon 
one  end  of  an  “original  line,”  determined  to  be  such  by  old 
“marks,”  or  by  testimony  as  to  its  having  been  undisputed 
for  twenty  years  or  more,  and  turn  the  vernier  plate  till  the 
reading  is  the  same  as  that  given  in  the  deed ; then  run 
the  other  lines  by  the  bearings  and  distances  as  given  in 
the  deed. 

174.  To  make  allowance  for  change  in  “variation”  in 
running  out  old  lines,  when  only  one  “corner”  can  be  identi- 
fied, the  surveyor  must  see  to  it  that  the  date  of  the  deed 
from  which  he  is  working  is  the  same  as  the  date  of  the  survey 
from  which  the  description  of  property  in  the  deed  is  taken  ; 
very  often  descriptions  are  copied  from  former  deeds,  in 
which  case  the  date  of  the  survey  must  be  discovered  before 
running  the  lines. 

175.  If  no  “ original  ” line  of  the  property  can  be  found, 
nor  a corner  and  date  of  survey,  then  the  surveyor  must  seek 
his  data  upon  neighboring  property  and  work  back  from  there, 
through  several  deeds  if  necessary. 

176.  It  is  very  desirable  that  the  date  of  the  survey  from 


152 


ELEMENTS  OF  SURVEYING. 


[HOOK  III. 


which  a description  is  taken,  and  also  the  name  of  the  surveyor, 
should  be  entered  in  every  deed  conveying  lands. 

The  form  of  a “ Survey  Bill ” may  be  somewhat  as  follows  : 

Description  of  Property  in  Town  of  , County  of 

, State  of  — , surveyed  on  the  sixteenth  day  of 

August,  in  the  year  eighteen  hundred  and  eighty-three,  and 
bounded  as  follows  : 

Beginning  at  a point  in  the  northwest  corner  of  land  owned 
by  A.  B.  C.,  and  marked  by  a cross  cut  into  the  top  of  a stone 
sunk  three  feet  into  the  ground, — from  which  point  the  east 
chimney  of  the  house  of  A.  B.  C.  bore  S 87£°  E,  distant  8.50  eh. — 
thence  from  the  point  of  beginning  N 18°  E,  11.27  ch.  to  a 
stake  ; — thence  N 56°  E,  15.26  ch.  to  a corner  where  two  stone 
walls  meet; — thence  S 87°  E,  5.56  ch.  to  the  shore  line  of  the 
bay ; — thence  following  the  shore  line  to  a point  on  the  shore 
line  bearing  S 23°  W,  and  distant  in  a straight  line  13.47  ch. — 
from  which  point  the  middle  of  the  eastern  key-stone  of  the 
bridge  across  Beaver  Dam  Creek  bears  S 37£°  E ; — thence,  fol- 
lowing the  middle  of  the  road  to  a point  in  the  middle  of  the 
road  bearing  S 87°  W,  distant  in  a straight  line  5 ch. — thence, 
....&c.,  &c. ; the  above  described  property  containing,  by  cal- 
culation, — Acres,  — Roods,  and  — Perches,  more  or  less. 

Erasmus  Ruggles, 

Variation  of  the  needle  — ° — ' — " W (or  E).  Surveyor. 


BOOK  IV. 

TRANSIT  SURVEYING. 


SECTION  I. 

SU  RVEYO  R'S  TRANSIT. 

177.  The  transit  is  an  instrument  used  for  measuring  hori- 
zontal and  (when  furnished  with  a vertical  circle)  vertical  angles. 
It  is  placed  on  a tripod,  TTT,  to  which  it  is  screwed  fast  by 
means  of  a horizontal  brass  plate,  DE.  Upon  the  upper  surface 
of  the  plate  DE  rest  four  screws  with  milled  heads,  called  level- 
ing screws,  which  work  through  the  second  horizontal  plate,  EG, 
into  cylindrical  nuts,  shown  in  the  figure.  The  two  plates  DE 
and  EG  are  called  leveling  plates.  The  lower  leveling  plate,  DE, 
is  made  in  two  pieces — the  upper  piece,  which  is  screwed  fast  to 
the  top  of.  the  tripod,  having  a large  opening  in  its  centre,  in 
which  the  smaller  lower  one  is  shifted  from  side  to  side,  or 
turned  completely  around. 

By  this  arrangement,  termed  a “ shifting  centre,”  the  instru- 
ment is  easily  moved  over  the  upper  plate,  and  the  plummet, 
which  hangs  from  the  centre,  set  precisely  over  a point  without 
moving  the  tripod. 

The  upper  side  of  the  plate  DE  terminates  in  a curved 
surface,  which  encloses  a ball  that  is  nearly  a hemisphere  with 
the  plane  of  its  base  horizontal.  This  ball  or  hemispherical  nut 
is  screwed  fast  to  the  smaller  base  of  a solid  conic  spindle,  that 
passes  through  the  curved  part  of  the  plate  DE.  To  this 
spindle  is  firmly  attached  the  second  horizontal  plate,  EG. 


154 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


To  this  spindle,  also,  and  above  the  plate  FG , is  fitted  and 
fastened  by  spring-catch,  a socket,  called  the  main  socket,  to 
which  is  attached  a third  horizontal  and  circular  plate,  J3B , 


Fio.  85. 


called  the  limb  of  the  instrument.  Fitted  to  the  upper 
conical  surface  of  the  main  socket  is  a second  socket  to  which 
is  united  a thin  circular  plate,  A A,  called  the  Vernier  Plate, 


SEC.  I.J 


surveyor’s  transit. 


155 


which  rests  on  the  limb  of  the  instrument  and  carries  a compass- 
circle,  standards,  &c.,  as  shown  in  Fig.  85. 


Fig.  86. 


178.  The  sectional  view,  Fig.  86,  shows  the  interior  construc- 
tion of  the  sockets  of  the  transit,  the  manner  in  which  it  is 
detached  from  the  spindle,  and  the  means  by  which  it  can  be 
taken  apart  if  desired. 

In  the  figure,  the  limb  BB  is  attached  to  the  main  socket  of 
LL , which  is  itself  carefully  fitted  to  the  conical  spindle  P,  and 
held  in  place  by  the  spring  catch  C. 

The  upper  plate,  A A,  carrying  the  compass-circle,  standards, 
&c.,  is  fastened  to  the  flanges  of  the  socket,  UU,  which  is  fitted 
to  the  upper  conical  surface  of  the  main  socket  L ; the  weight  of 
all  the  parts  being  supported  on  the  small  bearings  of  the  end  of 
the  socket,  as  shown,  so  as  to  turn  with  the  least  possible 
friction. 

A small  conical  centre,  in  which  from  below  is  inserted  the 
strong  screw,  8,  is  brought  down  firmly  upon  the  upper  end  of 
the  main  socket  LL,  and  thus  holds  the  two  .plates  of  the  instru- 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


156 

ment  securely  together,  while  at  the  same  time  allowing  them  to 
move  freely  around  each  other  in  use. 

A small  disc  above  the  conical  centre  contains  the  steel 
centre-pin  upon  which  rests  the  needle,  as  shown  ; the  disc  is 
fastened  to  the  upper  plate  by  two  small  screws,  as  represented. 

The  main  socket  with  all  its  parts  is  of  the  best  bell-metal 
and  is  most  carefully  and  thoroughly  made,  the  long  bearing  of 
the  sockets  ensuring  their  tirm  and  easy  movement,  while  at  the 
same  time  they  are  entirely  out  of  the  reach  of  dust,  or  other 
source  of  wear. 

When  desired  the  whole  upper  part  of  the  instrument  can  be 
taken  off  from  the  spindle  by  pulling  out  the  head  of  the  spring- 
catch  at  C,  and  when  replaced  will  be  secured  by  the  self-acting 
spring  of  the  catch. 

The  figure  also  shows  the  covers  of  the  leveling  screws,  the 
shifting  centre  of  the  lower  leveling  plate,  and  the  screw  and 
loop  for  the  attachment  of  the  plummet. 

179.  On  the  upper  surface  of  the  plate  FG,  Fig.  85,  rests  a 
clamp  which  goes  round  the  main  socket  and  which,  being  com- 
pressed by  the  clamp-screw  K , is  made  fast  to  it.  This  clamp  is 
thus  connected  with  the  plate  FG.  Two  small  cylinders,  act,  are 
fastened  to  the  plate  FG ; through  these  cylinders  thumb 
screws,  II,  called  slow-motion  screws  or  tangent  screws,  pass  and 
abut  against  opposite  sides  of  a piece  projecting  from  the  clamp- 
ring, thus  preventing  the  clamp  from  moving  in  either  direction. 
When  the  clamp  is  compressed  against  the  main  socket  by  the 
clamp-screw  K,  the  limb  of  the  instrument  is  prevented  from 
revolving.  One  of  the  tangent  screws  I,  being  then  loosened  as 
the  other  is  tightened,  will  slowly  and  steadily  move  the  clamp- 
ring itself  and  with  it,  of  course,  the  limb. 

The  limb,  BB,  has  a silvered  circle  near  its  outer  edge,  on 
which  tho  graduation  for  horizontal  angles  is  made.  In  the 


SEC.  I.] 


SURVEYOR’S  TRANSIT. 


157 


instrument  described,  the  circle  is  divided  into  degrees  and  half 
degrees ; the  degrees  are  figured  in  two  rows,  viz : from  0 to  360, 
and  from  0 to  90  each  way. 

The  vernier  plate,  AA,  has  two  openings,  one  of  which  is 
shown  in  the  Figure  85,  placed  diametrically  opposite  each  other, 
in  which,  attached  to  the  plate  ^4^4  and  moving  with  it,  are 
small  silvered  arcs  called  verniers,  which  serve  to  read  the  limb 
about  which  they  revolve. 

The  verniers  are  double,  having  on  each  side  of  the  zero 
mark  thirty  equal  divisions  corresponding  precisely  with  twenty- 
nine  half-degrees  of  the  limb  ; they  thus  read  to  single  minutes, 
and  the  number  passed  over  is  counted  in  the  same  direction  in 
which  the  vernier  is  moved. 

The  use  of  two  opposite  verniers  gives  the  means  of  testing 
the  correctness  of  the  graduations,  the  perfection  with  which 
they  are  centered  and  the  dependence  which  can  be  placed  upon 
the  accuracy  of  the  angles  indicated. 

Two  spirit  levels,  at  right  angles  to  each  other,  are  attached 
to  the  vernier  plate  by  small  adjusting  screws,  one  being  fixed 
to  the  standard  which  supports  the  telescope,  so  as  not  to  obstruct 
the  light  which  falls  on  the  vernier  opening  beneath. 

The  vernier  plate  turns  freely  around  with  the  socket  to 
which  it  is  attached.  It  is  made  fast  to  the  limb  by  the  clamp- 
screw  Q,  Fig.  85,  after  which  the  smaller  motions  are  made  by 
the  tangent-screw  R. 

There  is  a compass  on  the  vernier  plate  that  is  concentric 
with  it,  the  uses  of  which  have  been  explained  under  the  head 
Compass.  It  also  may  be  made  to  serve  as  a check  upon  the 
measurement  of  angles  with  the  transit,  as  from  the  magnetic 
bearings  of  two  courses  may  be  found  the  angle  between  them,  as 
explained  in  Art.  114. 

180.  The  standards  which  support  the  horizontal  axes  of  the 


158 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


telescope  with  its  attached  level,  rest  on  and  are  made  fast  to  the 
vernier  plate.  The  vertical  circle  MM , Fig.  85,  called  the 
Vertical  Limb,  is  securely  fastened  to 
the  axis  of  the  telescope;  it  is  plated 
with  silver,  graduated  to  half  degrees, 
and  with  its  vernier  enables  the  sur- 
veyor to  obtain  vertical  angles  to  single 
minutes.  There  is  a clamp-and-tangent 
arrangement  connected  with  the  tele- 
scope ; it  consists  of  an  arm  at  one  end 
encircling  the  telescope  axis,  and  at  the 
other  connected  with  the  tangent-screw 
0,  Fig.  85.  The  clamp  is  fastened  at 
will  to  the  axis  by  a clamp-screw,  and 
then  by  turning  the  tangent-screw  the 
telescope  is  raised  or  lowered  as  desired. 


181.  The  telescope  is  from  ten  to 
eleven  inches  long,  firmly  secured  to  an 
axis  having  its  bearings  nicely  fitted  in 
the  standards,  and  thus  enabling  the 
telescope  to  be  moved  in  either  direction, 
or  turned  completely  around  if  desired. 

The  different  parts  of  the  telescope 
are  shown  in  Fig.  87. 

The  object-glass  is  composed  of  two 
lenses,  so  as  to  show  objects  without 
color  or  distortion,  is  placed  at  the  end 
of  a slide  having  two  bearings,  one  at 
the  end  of  the  outer  tube,  the  other  in 
the  ring  GO,  suspended  within  the  tube 
by  four  screws,  only  two  of  which  are 
shown  in  the  cut. 


SEC.  I.J 


SURVEYOR’S  TRANSIT. 


159 


The  object-glass  is  carried  out  or  in  by  a pinion  working 
in  a rack  attached  to  the  slide,  and  thus  adjusted  to  objects 
either  near  or  remote  as  desired. 

The  eye-piece  is  made  up  of  four  plane  convex  lenses,  which, 
beginning  at  the  eye  end,  are  called  respectively  the  eye,  the 
field,  the  amplifying,  and  the  object  lenses,  the  whole  forming  a 
compound  microscope  having  its  focus  in  the  plane  of  the  cross- 
wire ring  BB. 

The  eye-piece  is  brought  to  its  proper  focus  usually  by 
twisting  its  milled  end,  the  spiral  movement  within  carrying  the 
eye-tube  out  or  in  as  desired  ; sometimes  a pinion,  like  that 
which  focuses  the  object-glass,  is  employed  for  the  same  purpose. 

182.  In  order  that  the  telescope  may  be  directed  to  an  object 
with  precision,  two  spider’s  lines,  or  fine  wires,  are  fixed  at  right 
angles  to  each  other,  and  placed  within  the  barrel  of  the  tele- 
scope, and  at  the  focus  of  the  eye-glass. 

The  cross-wire  diaphragm  is  a 
small  ring  of  metal  suspended  in 
the  tube  of  the  telescope  by  four 
capstan-head  screws,  ff,  gg,  Fig.  88. 

The  ring  can  thus  be  moved  in 
either  direction  by  working  the 
screws  with  an  adjusting-pin. 

Across  the  flat  surface  of  the  ring 
two  fine  fibres  of  spider’s  web,  or, 
better,  very  fine  platinum- wire,  are 
extended  at  right  angles  to  each  other,  their  ends  being  cemented 
into  fine  lines  cut  in  the  metal  of  the  ring. 

The  intersection  of  the  wires  forms  a very  minute  point 
whicn,  when  they  are  adjusted,  determines  the  optical  axis  of  the 
telescope,  and  enables  the  surveyor  to  fix  it  upon  an  object  with 
the  greatest  precision. 


Fig.  88. 


160 


ELEMENTS  OF  SURVEYING. 


[ROOK  IV. 


The  openings  in  the  telescope  tube  are  made  considerably 
larger  than  the  screws,  so  that,  when  these  are  loosened,  the  ring 
may  be  moved  a short  distance  in  the  direction  of  the  length  of 
either  wire  for  purposes  of  adjustment  as  hereafter  described,  and 
may  also  be  slightly  turned  for  the  same  purpose. 

183.  To  measure  horizontal  angles  correctly,  the  limb  of 
the  instrument  must  be  made  truly  horizontal. 

To  level  the  instrument,  extend  the  legs  and  place  them  so  as 
to  bring  each  bubble  as  nearly  as  possible  to  the  middle  of  its 
tube.  Turn  the  vernier  plate  till  one  of  the  levels  is  parallel  to 
one  pair  of  leveling  screws,  the  other  level  will  be  parallel  to  the 
other  pair  of  screws.  If  the  bubble  in  either  level  is  not  at  the 
middle,  turn  the  leveling  screws  to  which  that  level  is  parallel,  in 
contrary  directions,  thus  raising  one  side  of  the  horizontal  plate 
carrying  the  graduated  circle  and  lowering  the  other  till  the 
bubble  is  brought  to  the  middle  of  that  tube ; with  the  other 
pair  of  leveling  screws,  bring  the  bubble  in  the  other  level  to  the 
middle  of  the  tube;  when  the  bubbles  in  both  levels  are  at  the 
middle  of  their  respective  tubes,  the  plane  of  the  levels,  and, 
consequently,  the  graduated  circle  by  which  horizontal  angles  are 
measured,  are  truly  horizontal.  Two  opposite  leveling  screws 
may  be  turned  in  contrary  directions,  by  holding  each  screw  with 
the  thumb  and  forefinger,  and  turning  so  that  both  thumbs  turn 
in,  or  both  out.  When  the  thumbs  are  turned  outward,  the  left- 
hand  side  of  the  circle  is  raised;  when  both  are  turned  inward, 
the  right-hand  side  is  raised.  The  bubble  always  runs  to  the  end 
of  the  level  which  is  too  high. 

If  the  leveling  screws  become  jammed,  so  as  to  work  hard, 
turn  one  of  them  only,  forward  or  backward,  till  they  are  free 
again  ; sometimes  they  work  hard,  from  setting  the  other  pair 
tight  while  at  a large  angle  of  inclination  with  the  lower  plate, 
in  which  case  the  other  pair  must  be  loosened  again. 


SURVEYOR'S  TRANSIT. 


161 


V AA 

sec.  i.]  yJ 

ADJUSTMENTS  OF  THE  TRANSIT. 

/ Before  using  the  instrument,  it  must  be  adj listed ; that 
is,  the  parts  must  be  brought  to  their  proper  relative  positions. 
There  are  four  principal  adjustments. 

First  Adjustment.— To  make  the  axes  of  the  levels , on  the 
limb,  'perpendicular  to  the  axis  of  the  instrument. 

184.  Turn  the  horizontal  limb  until  one  of  the  levels  is 
parallel  to  one  pair  of  leveling  screws  ; the  other  will  be  parallel 
to  the  other  pair.  Bring  the  bubble  in  each  tube  to  the  middle 
with  its  parallel  pair  of  leveling  screws ; after  which  turn  the 
horizontal  limb  half  way  round  to  reverse  the  levels,  and  if  the 
bubbles  remain  in  the  middle  of  the  tubes,  the  levels  are  properly 
adjusted.  But  if  either  bubble  recedes  from  the  centre,  that 
level  must  be  adjusted  by  raising  the  lower  or  depressing  the 
higher  end,  one  half  by  the  leveling  screws,  and  one  half  by  the 
screws  which  fasten  the  level  to  the  plate.  Again  reverse  the 
levels,  and  if  the  bubble  does  not  remain  in  the  middle  of  the 
tube,  correct  the  error  as  before,  and  repeat  the  operation  till  the 
bubble  remains  in  the  middle  during  an  entire  revolution  of  the 
plate. 

Each  level  must  be  adjusted  separately. 

Second  Adjustment. — To  fix  the  intersection  of  the  cross- 
wires in  the  axis  of  the  telescope , which  is  called  the  line  of 
collimation. 

185.  Having  screwed  the  tripod  to  the  instrument,  extend 
the  legs,  and  place  them  firmly.  Level  carefully,  then  loosen  the 
clamp-screw  Q,  of  the  vernier  plate,  and  direct  the  telescope  to  a 
small,  well-defined,  and  distant  object.  Slide  the  eye-glass  of  the 
telescope  till  the  cross-wires  are  distinctly  seen ; then  with  the 


162 


ELEMENTS  OF  SURVEYING. 


[ROOK  IV. 


thumb-screw,  which  forces  out  and  draws  in  the  object-glass, 
adjust  this  glass  to  its  proper  focus,  when  the  object,  as  well  as 
the  cross-wires,  will  be  distinctly  seen  ; clamp  the  vernier  plate 
and,  by  the  tangent-screw  R,  bring  the  intersection  of  the  cross- 
wires exactly  upon  a well-defined  point  of  the  object. 

Now  move  the  eye  from  side  to  side  across  the  eye-glass ; if 
there  results  any  movement  of  the  cross- wires  away  from  the 
point  sighted,  it  is  evidence  that  the  image  of  the  point  does  not 
fall  exactly  upon  the  plane  of  the  cross-wires,  and  the  object-glass 
must  be  again  focussed,  till  no  such  displacement  can  be  detected. 
This  displacement  is  called  “ parallax.” 

Having  done  this,  sight  some  well-defined -point,  o*  the  mid- 
dle of  a pin,  distant  two  or  three  hundred  feet,  and  having 
clamped  the  horizontal  motions,  revolve  the  telescope  about  its 
horizontal  axis  and  sight  a pin  in  the  opposite  direction  from  the 
first  and  at  the  same  distance  from  the  plumb;  now  loosen  the 
lower  clamp  and,  revolving  the  instrument  about  the  vertical 
axis,  sight  the  middle  of  the  first  pin  again,  and  clamp ; reverse 
the  telescope  upon  the  horizontal  axis,  and  if  it  now  cuts  the 
second  pin,  as  before,  the  adjustment  is  correct;  but  if  the 
intersection  of  the  wires  falls  on  one  side  of  the  second  pin, 
bring  it  back  over  one  quarter  of  the  error,  by  the  capstan  screws 
on  each  side  of  the  telescope  over  the  cross-wire  ring.  As  the 
eye-piece  inverts  the  position  of  the  wires,  the  operation  of  loos- 
ening one  of  the  screws  and  tightening  the  other  on  the  opposite 
side,  must  be  conducted  as  if  to  increase  the  error  observed. 
To  insure  the  accuracy  of  the  adjustment,  repeat  the  whole 
operation  as  above  described,  until  the  reversals  cut  the  pins 
exaptly. 

186.  The  reason  for  the  above  directions  will  be  apparent 
upon  examination  of  Tig.  89. 


SEC.  I.] 


SURVEYOR’S  TRANSIT. 


163 


Let  A A'  represent  the 
line  of  collimation  pro- 
longed both  ways  from  the 
instrument  at  I,  and  sup- 
pose that  the  intersection 
falls  at  B,  the  cross- wires  not  being  in  adjustment;  on  reversing 
the  telescope  the  intersection  will  fall  at  B',  as  far  to  the  left  of 
A’  as  B was  to  the  right  of  A;  if  now  we  turn  the  instrument 
180°,  by  the  horizontal  limb,  the  intersection  will  fall  at  C,  to 
the  left  of  A,  CA  being  equal  to  BA , and  on  reversal  the  inter- 
section will  fall  at  C,  CA ' being  equal  to  A'Br ; if  the  cross- 
wires be  moved  over  CA',  the  adjustment  will  be  accomplished. 
But  it  is  not  well  to  risk  introducing  possible  errors  of  gradua- 
tion of  the  horizontal  limb,  or  errors  of  reading,  into  the  opera- 
tion, and  therefore  instead  of  turning  the  instrument  through 
180°,  the  surveyor  turns  till  he  sights  B the  second  time,  thus 
passing  over  180°  ± CB  ; this  causes  C to  fall  at  G",  CO"  being 
equal  to  CB,  and  therefore  B'C"  must  be  four  times  CA',  or 
four  times  the  error  of  adjustment. 

187.  The  adjustment  of  the  horizontal  wire,  which  is  only  an 
approximation  to  perfect  adjustment,  is  accomplished  thus : 
Set  the  zero  of  the  vertical  limb  at  the  zero  of  its  vernier,  and 
clamp;  sight  some  sharply  defined  point  upon  a wall  or  staff, 
distant  about  200  feet ; unclamp  and  reverse  the  telescope  upon 
its  horizontal  axis  and  set  the  zero  of  the  vernier  at  180°  on  the 
vertical  limb  ; now  unclamp  the  lower  clamp  and  revolve  the 
instrument  upon  its  vertical  axis  till  the  chosen  point  is  again 
in  the  field  of  view ; if  the  point  and  the  intersection  of  the 
cross-wires  do  not  coincide,  correct  half  the  error  by  the  screws, 
moving  the  cross-wire  ring  up  or  down,  and  the  remaining  half 
error  by  the  leveling  screws ; repeat  the  operations  till  the  error 
is  wholly  corrected. 


164 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


If  the  instrument  should  not  be  furnished  with  a vertical 
limb,  the  adjustment  of  the  horizontal  hair  can  only  be  guessed 
at,  by  bringing  it  to  the  centre  of  the  field  of  the  telescope. 

Third  Adjustment.— 7b  wake  the  axis  of  the  attached  level 
of  the  telescope  parallel  to  the  line  of  collimation. 

188.  First  level  the  instrument  carefully,  and  with  the  clamp 
and  tangent  movement  to  the  axis,  make  the  telescope  horizontal 
as  near  as  may  be  with  the  eye  ; then  having  the  line  of  collima- 
tion previously  adjusted,  drive  a stake  at  a convenient  distance, 
say  from  one  to  two  hundred  feet,  and  note  the  height  cut  by 
the  horizontal  cross-wire  upon  a staff  set  on  the  top  of  the  stake. 
Fix  another  stake  in  the  opposite  direction  and  at  the  same  dis- 
tance from  the  instrument,  and,  without  disturbing  the  telescope, 
turn  the  instrument  upon  its  spindle,  set  the  staff  upon  the 
stake,  and  drive  in  the  ground  until  the  same  height  is  indicated 
as  in  the  first  observation.  The  top  of  the  two  stakes  will  then 
be  in  the  same  horizontal  line,  however  much  the  telescope  may 
be  out  of  level. 

Now  remove  the  instrument  from  fifty  to  one  hundred  feet  to 
one  side  of  either  of  the  stakes,  and  in  line  with  both ; again  level 
the  instrument,  clamp  the  telescope  as  nearly  horizontal  as  may 
be,  and  note  the  heights  indicated  upon  the  staff  placed  first 
upon  the  nearer  and  then  upon  the  more  distant  stake. 

If  both  readings  do  not  agree,  correct  nearly  the  whole  error 
of  the  more  distant  reading  by  the  tangent  screw,  and  continue 
correcting  and  reading  until  the  two  readings  agree  ; now  place 
a stake  at  a point  in  line  with  the  two  already  fixed  and  about 
one  hundred  feet  beyond  the  more  remote,  and  mark  upon  it  the 
point  cut  by  the  cross-wires,  and  mark  also  the  corresponding 
point  upon  the  more  distant  of  the  first  two  stakes ; a line 
passing  through  these  two  marks  will  be  a horizontal  or  level 


SEC.  I.] 


SURVEYOR’S  TRANSIT. 


165 


line  with  reference  to  the  first  position  of  the  instrument,  but 
not  with  reference  to  its  present  position. 

Set  the  instrument  again  at  its  first  position,  and  by  the 
tangent-screw  make  the  readings,  above  or  below  the  marks,  the 
same;  the  line  of  collimation  will  now  be  horizontal,  and  the 
bubble  of  the  attached  level  must  be  brought,  very  carefully,  to 
the  middle  of  its  run  by  the  screws  at  the  end  of  the  level  tube. 

Fourth  Adjustment. — To  make  the  axis  of  the  vertical  limb 
perpendicular  to  the  axis  of  the  instrument. 

189.  Bring  the  intersection  of  the  cross-wires  of  the  tele- 
scope upon  a plumb-line,  or  any  well-defined  vertical  object, 
and  move  the  telescope  with  the  thumb-screw  0\  if  the  inter- 
section of  the  cross-wires  continues  on  the  vertical  line,  the  axis 
is  horizontal. 

Or,  the  adjustment  may  be  effected  thus : Direct  the  inter- 
section of  the  cross-wires  to  a well-defined  point  that  is  consider- 
ably elevated ; then  turn  the  vertical  limb,  until  the  cross-wires 
cut  some  other  well-defined  point,  upon  or  near  the  ground; 
revolve  the  telescope  on  its  axis,  and  turn  the  vernier  plate  180° ; 
then,  if  in  elevating  and  depressing  the  telescope  the  line  of 
collimation  passes  through  the  two  points  before  noted,  the  axis 
is  horizontal  If  it  be  found,  by  either  of  the  above  methods,  that 
the  axis  is  not  horizontal,  it  must  be  made  so  by  adjusting  the 
standards  which  support  the  telescope.  One  of  these  standards 
is  made  adjustable,  and  by  raising  the  standard  at  the  end  towards 
which  the  deviation  from  the  vertical  occurs,  or  depressing  the 
other,  the  adjustment  is  made. 

Note. — In  making  the  third  adjustment,  the  curvature  of  the  earth  is 
taken  into  account  that  the  adjustment  may  be  theoretically  correct. 


1G6 


ELEMENTS  OF  SURVEYING. 


[ROOK  IV. 


SECTION  II. 

MEASUREMENT  OF  ANGLES. 

190.  To  Measure  with  the  Transit  a Horizontal  Angle 
subtended  by  two  objects. — Place  the  axis  of  the  instrument 
directly  over  the  point  at  which  the  angle  is  to  be  measured. 
This  is  effected  by  means  of  a plumb,  suspended  from 
the  centre  of  the  plate  which  forms  the  upper  end  of  the 
tripod. 

Having  made  the  limb  truly  level,  place  the  0 of  the  vernier 
at  any  exact  degree  (merely  to  avoid  minutes  and  seconds  in  the 
first  reading),  and  fasten  the  clamp  screw  Q of  the  vernier  plate. 
Then,  facing  in  the  direction  between  the  lines  which  subtend 
the  angle  to  be  measured,  loosen  the  lower  clamp  and  sight  one 
of  the  objects  very  nearly,  without  wasting  time  in  trying  to 
secure  perfect  bisection  by  the  cross-wires ; tighten  the  lower 
clamp  and  make  perfect  bisection  by  the  lower  tangent  screw. 

This  being  done,  loosen  the  clamp-screw  Q of  the  vernier 
plate,  and  direct  the  telescope  to  the  other  object ; the  arc 
passed  over  by  the  0 point  of  the  vernier,  is  the  measure  of  the 
angle  sought ; the  difference  of  the  two  readings  (if  the  0°  of 
the  limb  be  not  passed  in  turning  to  the  second  object),  is  the 
required  angle  ; if  the  0°  be  passed  over,  then  add  360°  to  the 
smaller  reading  and  subtract  the  greater  reading  from  this  sum. 
Always  be  careful  to  take  both  readings  from  the  same  vernier. 

Note  1. — In  measuring  horizontal  angles,  it  does  not  matter 
whether  the  telescope  has  to  be  elevated  or  depressed  to  sight 
either  or  both  of  the  objects,  since  the  telescope  revolves  on  its 
axis  in  a vertical  plane,  and  the  angles  measured  arc  always  the 
horizontal  projections  of  the  angle. 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


167 


Note.  2. — When  great  accuracy  is  desired,  a repetition,  or 
several  repetitions  of  the  measure  may  be  made,  and  a mean  of 
the  observations  taken  as  the  true  measure.  (See  Arts.  254, 
255.) 

In  the  measurement  of  vertical  angles,  it  is  necessary  to 
understand,  first: 

191.  The  method  of  determining  the  index  error  of  the 
vertical  limb.  Having  leveled  the  horizontal  limb,  direct  the 
telescope  to  some  distinctly  marked  object,  as  the  top  of  a 
chimney,  and  read  the  instrument.  Revolve  the  telescope  on  its 
axis  and  turn  the  vernier  plate  180°,  and  having  directed 
the  telescope  to  the  same  object  again,  read  the  instrument. 
If  the  two  readings  are  the  same,  the  limb  is  adjusted  ; that  is, 
the  0 of  the  limb  coincides  with  the  0 of  its  vernier,  when  the 
axis  of  the  telescope  is  parallel  to  the  horizontal  limb. 

When  the  reading,  found  with  the  telescope  in  the  first 
position,  is  greater  than  that  obtained  in  the  reversed  position, 
the  true  elevation  of  the  object,  which  is  equal  to  a mean 
of  the  readings,  may  be  obtained  by  subtracting  half  the 
difference  from  the  first  reading.  If  the  first  reading  is  less 
than  the  second,  the  half  difference  must  be  added  to  the 
first.  Hence, 

To  find  the  index  error,  take  the  reading  of  the  limb  when 
the  telescope  is  directed  to  a fixed  object,  and  then  with  the  telescope 
and  vernier  plate  both  reversed.  Take  half  the  difference  of  these 
readings,  and  affect  it  with  a minus  sign  if  the  first  is  the 
greater,  or  a plus  sign  if  the  second  is  the  greater  ; this  is  equal 
to  the  index  error. 

Let  the  operation  be  repeated  several  times,  using  different 
objects,  and  a mean  of  the  errors  will  be  more  correct  than  the 
result  of  a single  observation.  Then,  second: 

192.  Having  determined  the  index  error,  let  the  axis  of  the 


1(38 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


telescope  be  directed  to  any  point  either  above  or  below  the 
plane  of  the  limb,  and  read  the  arc  indicated  by  the  0 of  the 
vernier.  To  the  arc  so  read  apply  the  proper  correction,  if  any, 
and  the  result  will  be  the  true  angle  of  elevation  or  depression. 

The  angle  of  elevation  may  be  more  correctly  found  by 
taking  the  elevation  of  the  object,  and  repeating  the  observation 
with  the  telescope  and  vernier  plate  reversed,  and  then  taking 
a mean  of  the  readings  for  the  angle  required. 

193.  The  true  azimuth  of  a line  or  course  is  the  angle  which 
the  vertical  plane  through  it  makes  with  the  plane  of  the 
meridian. 

The  azimuth  of  a line  or  course  referred  to  some  preceding 
course,  or  to  any  given  line,  is  the  angle  made  by  the  line  or  course 
with  the  prolongation  of  the  line  of  reference  or  of  a parallel  to  it 
through  the  angular  point,  the  measurement  being  made  around 
to  the  right.  Thus, 

The  azimuth  of  BC  with  AB, 

(Fig.  90),  is  the  angle  RBC ; the 
azimuth  of  CD  with  AB  is  the  angle 
SCD  (SQ  being  parallel  to  RA). 

194.  To  find  with  the  transit 
the  azimuths  of  several  successive 
courses  with  a given  first  course. — 

Place  the  transit  at  B (Fig.  91)  and  level  it ; make  the  zero  of  the 
vernier  coincide  with  the  zero  of  the  horizontal  limb,  and  clamp 
the  vernier  plate ; direct  the 
telescope  to  A,  and  clamp 
the  limb;  revolve  the  tele- 
scope on  its  horizontal  axis, 
and  it  will  then  point  in 
the  direction  of  BR,  the 
prolongation  of  AB;  un- 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


169 


clamp  the  vernier  plate  and  direct  the  telescope  to  C;  the 
reading  will  be  the  angle  RBC,  the  azimuth  of  BC  with  AB. 

Clamp  the  vernier  plate  and  remove  the  instrument  to  G; 
reverse  the  telescope  on  its  horizontal  axis,  loosen  the  lower 
clamp  and  sight  B ; the  horizontal  limb  now  has  its  zero  point 
in  the  direction  of  QP , or  its  parallel  AB,  as  it  had  at  B ; 
tighten  the  lower  clamp  and  revolve  the  telescope  on  its  axis ; 
unclamp  the  vernier  plate,  direct  the  telescope  to  D,  and  the 
reading  will  be  the  angle  QCD,  which  CD  makes  with  PQ,  or  its 
parallel  AB,  and  is  the  azimuth  of  CD  with  AB. 

Clamp  the  vernier  plate  and  remove  the  transit  to  D ; reverse 
the  telescope  on  its  horizontal  axis,  loosen  the  lower  clamp  and 
sight  C ; the  limb  will  then  have  its  zero  point  in  the  direction 
TS,  or  its  parallel  AB,  as  it  had  at  C and  B ; tighten  the  lower 
clamp  and  revolve  the  telescope  on  its  axis  ; unclamp  the  vernier 
plate  and  direct  the  telescope  to  E ; the  reading  will  be  the 
angle  TDE,  which  DE  makes  with  TS,  or  its  parallel  BA,  and  is 
the  azimuth  of  DE  with  AB. 

Proceed  in  like  manner  with  any  number  of  successive 
courses. 

If  the  courses  enclose  a field,  the  reading  at  the  last  station, 
sighting  to  the  first  station  occupied  by  the  transit,  should  be  360°. 

The  course  AB,  with  respect  to  which  the  azimuths  are 
taken,  is  called  the  Meridian  of  the  Survey. 

195.  The  magnetic  bearing  of  a line  or  course,  is  the  angle 
which  it  makes  with  the  magnetic  meridian,  and  its  true  bearing 
is  the  angle  which  it  makes  with  the  true  meridian. 

In  finding  the  area  of  a piece  of  ground,  it  is  not  necessary  to 
have  either  the  true  or  the  magnetic  bearing.  It  is  sufficient  to 
have  the  bearings  of  the  several  successive  courses  with  respect  to 
one  of  the  courses  taken  as  a meridian.  These  may  be  found 
from  the  azimuths  as  follows: 


170 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


First  suppose  a north  and  south  line,  and 
an  east  and  west  line  to  be  drawn,  and  the 
graduation  to  he  made  from  0°  to  3G0°, 
as  represented  in  Fig.  92  ; let  the  course 
taken  as  the  meridian  be  represented  by  the 
line  NS;  then  when  the  azimuth  of  the  fig.  92. 

second  course  with  the  first,  or  meridian,  is  less  than  90°,  it  is 
the  bearing,  and  since  the  course  lies  between  N and  E , the 
bearing  is  NE ; when  the  azimuth  is  90°,  the  bearing  is  due 
east ; when  the  azimuth  is  between  90°  and  180°,  the  course 
lies  between  S and  E , the  bearing  with  the  first  course,  or 
meridian,  will  be  SE,  and  may  be  obtained  by  subtracting  the 
azimuth  from  180°  ; when  the  azimuth  is  180°,  the  bearing  is  due 
south  ; when  the  azimuth  is  between  180°  and  270°,  the  course 
lies  between  S and  W,  the  bearing  is,  therefore,  SW , and  may 
be  obtained  by  subtracting  180°  from  the  azimuth;  when 
the  azimuth  is  270°,  the  bearing  is  due  west ; when  the 
azimuth  is  between  270°  and  360°,  the 
course  lies  between  N and  W,  the  bear- 
ing is  NW,  and  may  be  obtained  by 
subtracting  the  azimuth  from  360°. 

For  example:  Let  AB  (Fig.  93)  be 
taken  as  the  meridian,  and  let  the 
azimuths  of  the  several  courses  with 
AB  be  as  in  the  table ; then  will  the 
bearings  of  the  several  courses  with 
AB , be  as  noted  in  the 

TABLE. 


8tation. 

Azimuth  with  AB. 

Bearing  with  AB. 

A 

0° 

North 

B 

93°  30' 

S 86°  30'  E 

C 

175°  30' 

S 4°  30'  E 

I) 

257° 

s 7 r w 

c 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


m 


196.  If  it  is  desired  to  find  the  true  or  magnetic  bearing  of  a 
course  from  its  azimuth  with  a given  course  taken  as  a meridian, 
it  may  be  obtained  by  finding  the  true  or  magnetic  bearing  of 
the  course  taken  as  meridian  and  subtracting  it  from,  or  adding  it 
to,  the  azimuth  of  the  given  course,  according  as  the  bearing  of 
the  reference  line  is  NW  or  NE  ; the  result  if  less  than  90°  is 
the  true,  or  magnetic  bearing,  and  is  NE ; if  more  than  90°  and 
less  than  180°,  subtract  it  from  180°,  and  the  result  will  be  the 
bearing,  SE ; &c. 

For  illustration,  take  the  example  in  the  last  article,  and  let 
the  magnetic  bearing  of  the  meridian  course,  or  reference  line, 
AB,  be  N 31|  W. 

TABLE. 


Station. 

Azimuth  with  AB. 

Bearing  with  AB. 

Magnetic  Bearing. 

A 

0° 

North. 

N 31}  W 

B 

93°  30' 

S 86°  30'  E 

N 62°  E 

C 

175°  30' 

S 4°  30'  E . 

S 36°  E 

D 

257° 

S 77°  W 

S 45°  30'  W 

If  BC,  of  which  the  magnetic  bearing  is  N 62°  E,  had  been 
taken  as  the  reference  line,  its  bearing  would  have  been  added  to 
the  azimuths  to  obtain  tha  magnetic  bearings  of  the  successive 
courses,  as  follows : 


Station. 

Azimuth  with  BC. 

Bearing  with  BC. 

Magnetic  Bearing. 

A 

226°  30' 

S 86°  30'  W 

N 31£  W 

B 

0° 

North. 

N 62°  E 

C 

o 

00 

N 82°  E 

S 36°  E 

D 

163°  30' 

S 10°  30'  E 

S 45|  W 

Note. — For  SE  and  SW  in  bearing  of  reference  line,  read 
NW  and  NE,  respectively,  in  applying  the  above  rule. 


m 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


To  find  the  True  Meridian  with  the  Transit. 

197.  Before  making  the  observations  it  will  be  necessary  to 
devise  some  means  by  which  the  cross-wires  may  be  lighted,  that 
they  may  be  distinctly  visible. 

To  do  this,  take  a board  of  about  one  foot  square,  paste 
white  paper  upon  it  and  perforate  it  through  the  centre ; the 
diameter  of  the  hole  being  somewhat  larger  than  the  diameter  of 
the  telescope  of  the  transit.  Let  this  board  be  so  fixed  to  a 
vertical  staff  as  to  slide  up  and  down  freely;  and  let  a small 
piece  of  board,  about  three  inches  square,  be  nailed  to  the 
lower  edge  of  it  for  the  purpose  of  holding  a candle. 

About  twenty-five  minutes  before  the  time  of  the  greatest 
eastern  or  western  elongation  of  the  pole-star,  as  shown  by  the 
table  of  elongations  (see  Art.  164),  let  the  transit  be  placed  at 
a convenient  point  and  leveled.  Let  the  board  be  placed  about 
one  foot  in  front  of  the  transit,  a lamp  or  candle  placed  on  the 
shelf  at  its  lower  edge,  and  let  the  board  be  slipped  up  or 
down  until  the  pole-star  can  be  seen  through  the  hole.  The 
light  reflected  from  the  paper  will  show  the  cross-wires  in  the 
telescope  of  the  transit. 

Then  let  the  vertical  cross- wire  be  brought  exactly  upon 
the  pole-star,  and,  if  it  is  an  eastern  elongation  that  is  to  be 
observed  and  the  star  has  not  yet  reached  the  most  east- 
erly point,  it  will  move  from  the  line  toward  the  east,  and  the 
reverse  when  the  elongation  is  west. 

At  the  time  the  star  attains  its  greatest  elongation,  it  will 
appear  to  coincide  'with  the  vertical  cross-wire  for  some 
time,  and  then  leave  it  in  the  direction  contrary  to  its  former 
motion. 

As  the  star  moves  toward  the  point  of  greatest  elongation, 
the  telescope  must  be  continually  directed  to  it  by  means  of 
the  tangent-screw  of  the  vernier  plate ; and  when  the  star  has 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


173 


attained  its  greatest  elongation,  great  care  should  be  taken  that 
the  instrument  be  not  afterward  moved. 

i 

Next  turn  the  telescope  very  carefully  upon  its  horizontal 
axis  and  fix  a peg  in  the  ground,  distant  150  or  200  feet ; to  do 
this,  let  the  light  of  the  lantern  shine  through  a small  hole  in  a 
board,  across  the  centre  of  which  hole  a plumb-line  hangs,  and 
by  sighting  the  line  thus  seen  the  peg  may  be  fixed.  Also  mark 
the  point  directly  under  the  transit  plumb  ; the  line  passing 
through  this  point  and  the  staff,  makes  an  angle  with  the  true 
meridian  equal  to  the  azimuth  of  the  pole-star. 

From  the  table  of  azimuths  (see  Art.  164),  take  the  azimuth 
corresponding  to  the  year  and  nearest  latitude.  If  the  observed 
elongation  was  east,  the  true  meridian  lies  on  the  west  of  the  line 
which  has  been  found,  and  makes  with  such  line  an  angle 
equal  to  the  azimuth.  If  the  elongation  was  west,  the  true 
meridian  lies  on  the  east  of  the  line  found;  and,  in  either 
case,  laying  off  the  azimuth  angle  with  the  transit,  gives  the 
true  meridian. 

198.  Another  method  depends  upon  the  fact  that  at  the 
same  angular  distances  east  and  west  of  the  meridian  the  alti- 
tudes of  any  selected  star  are  equal.  Direct  the  transit  to  any 
bright  star  towards  the  south,  and  east  of  the  meridian,  and 
clamp  the  vertical  limb ; carefully  read  the  horizontal  limb ; 
do  not  disturb  the  clamp  and  tangent  of  the  vertical  limb  in 
any  way,  but  loosen  the  clamp  of  the  vernier  plate,  and  after  a 
sufficient  interval  of  time,  two  or  three  hours  perhaps,  as  it  is 
not  good  practice  to  set  the  altitude  when  the  star  is  too  near 
the  meridian,  turn  the  plate  upon  its  spindle  towards  the  west ; 
bring  the  star  upon  the  cross-wire  intersection  as  its  altitude 
decreases,  moving  the  vernier  plate  only ; it  will  now  have  the 
same  altitude  as  before;  read  the  horizontal  limb  ; take  half  the 
difference  of  the  two  readings,,  and  set  this  half  angle  back 
towards  the  east ; the  line  thus  determined  is  the  meridian. 


174 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


APPLICATIONS  TO  HEIGHTS  AND  DISTANCES. 

199.  To  determine  the  horizontal  distance  to  a point 
which  is  inaccessible  by  reason  of  an  intervening  river. — 

Let  C be  the  point  (Fig.  94).  Measure 
along  the  bank  of  the  river  a hori- 
zontal base-line  AB,  and  select  the 
stations  A and  B,  in  such  a manner 
that  each  can  be  seen  from  the  other, 
and  the  point  C from  both  of  them. 

Then  measure  the  horizontal  angles 
CAB  and  CBA,  with  the  transit. 

Let  us  suppose  that  we  have  measured 

AB  = 600  yards  ; 

CAB  = A = 57°  35", 

and  CBA  = B = 64°  51'. 

Then,  C = 180°  - (A  + B)  = 57°  34'. 

To  find  the  distance  BC. 

sin  C : sin  A ::  AB  : BC. 

Applying  logarithms,  we  have, 

(a.  c.)  log  sin  C (57°  34')  ....  0.073649 

log  sin  A (57°  35')  ....  9.926431 

log  AB  (600)  2.778151 

log  BC  600.11  2.778231 

To  find  the  distance  AC. 

sin  C : sin  B ::  AB  : AC, 
and  applying  logarithms,  we  have, 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


175 


(a.  c.)  log  sin  C (57°  34/)  ....  0.073649 

log  sin  B (64°  51')  ....  9.956744 

log  AB  (600) 2.778151 

log  A C 643.94  2.808544 


To  determine  the  altitude  of  an  inaccessible  object  above  a 
given  horizontal  plane. 

FIRST  METHOD. 

200.  Suppose  D to  be  an  inac- 
cessible object,  and  BC  the  horizon- 
tal plane  from  which  the  altitude  is 
to  be  measured ; then,  if  we  suppose 
DC  to  be  a vertical  line,  it  will  repre- 
sent the  required  distance. 

Measure  any  horizontal  base-line, 
as  BA  ; and  at  the  extremities  B and  A , measure  the  horizontal 
angles  CBA  and  CAB.  Measure,  also,  the  angle  of  elevation 
DBC : 

Then,  in  the  triangle  CBA , there  will  be  known  two  angles 
and  the  side  AB ; the  side  BC  can  therefore  be  found  by 
calculation.  Having  found  BC,  we  shall  have,  in  the  right- 
angled  triangle  DBC , the  base  BC  and  the  angle  at  the  base, 
to  find  the  perpendicular  DC,  which  measures  the  altitude  of 
the  point  D above  the  horizontal  plane  BC. 

Let  us  suppose  that  we  have  found,  by  measurement, 

BA  = 780  yards. 

The  horizontal  angle  CBA  = B = 41°  24', 
the  horizontal  angle  CAB  = A = 96°  28', 
and  the  angle  of  elevation  DBC  = 10°  43'. 


D 


ELEMENTS  OF  SURVEYING. 


(ROOK  IV 


To  find,  in  the  triangle  BCA,  the  horizontal  distance  BC. 

The  angle  BCA  = C = 180°  — (A  + B)  = 42°  08'. 


Then,  sin  C : sin  A : : 

AB  : 

BO\ 

and  applying  logarithms,  we  have, 

(a.  c.)  log  sin  C (42°  08')  . . 

. 

0.173369 

log  sin  A (96°  28')  . . 

9.997228 

log  A B (780)  . . . 

2.892095 

log  BC  1155.29  yards  . 

3.062692 

In  the  right-angled  triangle  DCB, 

to  find  DC. 

We  have,  from  Theorem  IV, 

R : tan  DBG  : : 

BC  : 

DC. 

Applying  logarithms,  we  have, 

(a.  c. ) log  R (90°)  . . 

. 

0.000000 

log  tan  DBC  (10°  43')  . 

. 

9.277043 

log  BC  (1155.29)  . 

• • 

3.062692 

log  DC  218.64  . . 

• • 

2.339735 

Note  1. — It  might,  at  first,  appear,  that  the  solution  given 
requires  that  the  points  B and  A should  be  in  the  same 
horizontal  plane ; but  it  is  entirely  independent  of  such  a 
supposition. 

For,  the  horizontal  distance  represented  by  BA  is  the  same, 
whether  the  station  A is  on  the  same  level  with  B,  above  it, 
or  below  it.  The  horizontal  angles  CAB  and  CBA  are  also 
the  same,  so  long  as  the  point  C is  in  the  vertical  line  DC. 
Therefore,  if  the  horizontal  line  through  A should  cut  the  ver- 
tical line  DC,  at  any  point,  as  E,  above  or  below  C,  AB  would 
still  he  the  horizontal  distance  between  B and  A,  and  AE 
would  bo  the  horizontal  distance  between  A and  C. 


SEC.  II.] 


[EASUREMENT  OF  ANGLES. 


177 


If  at  A,  we  measure  the  angle  of  elevation  at  the  point  Z>, 
we  shall  know  in  the  right-angled  triangle  DAE,  the  base  AE 
and  the  angle  at  the  base  ; from  which  the  perpendicular  DE 
can  be  determined. 

Let  us  suppose  that  we  had  measured  the  angle  of  elevation 
DAE,  and  found  it  equal  to  20°  15'. 


First : In  the  triangle  BAC,  to  find  AC,  or  its  equal  AE. 

sin  C : sin  B ::  AB  : AC  or  AE. 


Applying  logarithms,  we  have, 

(a.  c.)  log  sin  C (42°  08')  ....  0.173369 

log  sin  B (41°  24')  ....  9.820406 

log  AB  (780) 2.892095 

log  AE  768.9  2.885870 


In  the  right-angled  triangle  DAE,  to  find  DE. 

We  have,  from  Theorem  IV, 

R : tan  A : : AE  : DE  : hence, 


(a.  c.)  log  R (90°) 0.000000 

log  tan  A (20°  15")  ....  9.566932 

log  AE  (768.9) 2.885870 

log  DE  283.66  2.452802 


Now,  since  DC  is  less  than  DE, 
it  follows  that  the  station  B is  above 
the  station  A.  That  is, 

DE  — DC  = 283.66  — 218.64  = 
65.02  = EC, 

which  expresses  the  vertical  distance 
that  the  station  B is  above  the  station  A 


178 


ELEMENTS  OF  SURVEYING. 


[ROOK  IV. 


Note  2. — It  should  be  remembered  that  the  vertical  distance, 
which  is  obtained  by  the  calculation,  is  estimated  from  a hori- 
zontal line  passing  through  the  eye,  at  the  time  of  observation. 
Hence,  the  height  of  the  instrument  is  to  be  added,  in  order  to 
obtain  the  true  result. 


SECOND  METHOD. 

201.  When  the  nature  of  the  ground  will  admit  of  it, 
measure  a base-line  AB,  in  the  direction  of  the  object  D.  Then 
measure,  with  the  instrument,  the  angles  of  elevation  at  A 
and  B. 


Then,  since  the  out- 
ward angle  DBC  is  equal 
to  the  sum  of  the  angles 
A and  ABB , it  follows 
that  the  angle  ADB  is 
equal  to  the  difference  of 

the  angles  of  elevation  at  A and  B.  Hence,  we  can  find  all  the 
parts  of  the  triangle  ADB.  Having  found  DB , and  knowing 
the  angle  DBC,  we  can  find  the  altitude  DC. 


This  method  supposes  that  the  stations  A and  B are  on  the 
same  horizontal  plane;  and  therefore  it  can  only  be  used  when 
the  line  AB  is  nearly  horizontal. 

Let  us  suppose  that  we  have  measured  the  base-line  and  the 
two  angles  of  elevation,  and  found, 


AB  = 975  yards,  A — 15°  3G\  and  DBC  = 27°  29'; 


required  the  altitude  DC. 


Ans.  DC  — 587.  G1  yards. 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


179 


To  determine  the  perpendicular  distance  of  an  object  below 
a given  horizontal  plane. 

202.  Suppose  G to  be  directly 
over  the  given  object,  and  A the 
point  through  which  the  horizontal 
plane  is  supposed  to  pass. 

Measure  a horizontal  base-line 

AB,  and  at  the  stations  A and  B 
conceive  the  two  horizontal  lines 

AC,  BC,  to  be  drawn.  The  oblique 
lines  from  A and  B,  to  the  object,  are  the  hypothenuses  of  two 
right-angled  triangles,  of  which  AC,  BC,  are  the  bases.  The 
perpendiculars  of  these  triangles  are  the  distances  from  the 
horizontal  lines  AC,  BC,  to  the  object.  If  we  turn  the  triangles 
about  their  bases  AC,  BC,  until  they  become  horizontal,  the 
object,  in  the  first  case,  will  fall  at  C' , and  in  the  second  at  C" . 

Measure  the  horizontal  angles  CAB,  CBA,  and  also  the 
angles  of  depression  C' AC,  C"BC. 

Suppose  that  we  have  measured,  and  found  AB  = 672  yards  ; 
BAC  — 72°  29';  ABC =39°  20';  angle  of  depression  C'AC  = 
27°  49',  and  C"BC=  19°  10'. 

First:  In  the  triangle  ABC,  the  horizontal  angle  ACB  = 
180°  — (A  -\-B)  = 180°  — 111°  49'  = 68°  11'. 

To  find  the  horizontal  distance  AC. 

sin  C : sin  B ::  AB  : AC;  hence. 


(a.  c.)  log  sin  C (68°  11') 0.032275 

log  sin  B (39°  20') 9.801973 

log  ^2?  (672) 2.827369 

log  AC  458.79  2.661617 


180 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


To  find  the  horizontal  distance  BC. 

sin  C : sin  A ::  AB  : BO ; whence, 

(a.  c.)  log  sin  C (68°  11') 0.032275 

log  sin  A (72°  29') 9.979380 

log  AB  (672) 2.827369 

log  BC  690.28  2.839024 

In  the  right-angled  triangle  CAC',  to  find  CC'. 

We  have.  Theorem  IV., 

R : tan  A : : AC  : CC';  whence, 

(a.  c.)  log  R (90°) 0.000000 

log  tan  A (27°  49') 9.722315 

log  A C 458.79  2.661617 

log  CC'  242.06  2.383932 

In  the  triangle  CBC",  to  find  CC". 

We  have,  Theorem  IV., 

R : tan  B ::  BC  : CC” ; whence, 


(a.  c.)  log  R (90°) 0.000000 

log  tan  B (19°  10') 9.541061 

log  BC  (690.28) 2.839024 

log  CC"  239.93  2.380085 

Hence,  also,  CC'  - CC"  = 242.06  - 239.93  = 2.13  yards; 


which  is  the  height  of  station  A above  station  B. 


SEC.  II. J 


measukement  oe  ahgles. 


181 


203. P R O B L E M S . 

1.  Wanting  to  know  the  distance  between  two  inaccessible 

objects,  which  lie  in  a direct  level  line  from  the  bottom  of  a 
tower  120  feet  in  height,  the  angles  of  depression  are  measured 
from  the  top  of  the  tower,  and  are  found  to  be,  of  the  nearer 
57°,  and  of  the  more  remote  25°  30'  ; required  the  distance, 
between  the  objects.  Ans.  173.656  feet. 

2.  In  order  to  find  the  distance  between 
two  trees,  A and  B,  which  could  not  be 
directly  measured  because  of  a pool  which 
occupied  the  intermediate  space,  the  dis- 
tances of  a third  point  C from  each  of  them 
were  measured,  and  also  the  included  angle 
A CB  ; it  was  found  that, 

CB  = 672  yards, 

CA  = 588  yards, 

A CB  = 55°  40'; 

required  the  distance  AB.  Ans.  592.967  yards. 

3.  Being  on  a horizontal  plane,  and  wanting  to  ascertain  the 
height  of  a tower  standing  on  the  top  of  an  inaccessible  hill, 
there  were  measured,  the  angle  of  elevation  of  the  top  of  the 
hill  40°,  and  of  the  top  of  the  tower  51°  ; then  measuring  in  a 
direct  line  180  feet  farther  from  the  hill,  the  angle  of  elevation 
of  the  top  of  the  tower  was  33°  45' : required  the  height  of  the 
tower. 

Ans.  83.998. 

4.  Wanting  to  know  the  horizontal  distance  between  two 
inaccessible  objects  E and  W,  the  following  measurements  were 
made ; 


182 


ELEMENTS  OF  SURVEYING. 


(BOOK  IV. 


AB  = 530  yards 
BAW  = 40°  16' 
viz.:  < WAE  = 57°  40' 
ABE  = 42°  22' 
EBW  = 71°  07'; 

required  the  distance  i?  W. 

Ans.  939.017  yards. 


Fio.  100. 


5.  Wanting  to  know  the 
horizontal  distance  between  two 
inaccessible  objects  A and  B , 
and  not  finding  any  station  from 
which  both  of  them  could  be 
seen,  two  points  C and  I)  were 
chosen  at  a distance  from  each 
other  equal  to  200  yards  ; from  the  former  of  these  points  A 
could  be  seen,  and  from  the  latter  B , and  at  each  of  the  points 
C and  D a staff  was  set  up.  From  C a distance  OF  was  measured, 
not  in  the  direction  DO , equal  to  200  yards,  and  from  D a 
distance  DE  equal  to  200  yards,  and  the  following  angles  taken. 


viz.: 


' AFC  = 83°  00', 
< ACD  = 53°  30', 
A OF  = 54°  31', 


BDE  = 54°  30', 
BDC  = 156°  25', 
BED  = 88°  30'. 


Ans.  AB  — 345.459  yards. 


0.  From  a station  P there  can  be  seen  three  objects,  A,  B , 
and  O,  whose  distances  from  each  other  are  known  : viz., 


A B = 800,  AO=  GOO,  and  BO  = 400  yards. 


Now,  there  are  measured  the  horizontal  angles, 

APO  = 33°  45',  and  liPC  = 22°  30' ; 


it  is  required  to  find  the  three  distances,  PA , PC,  and  PB. 


SEC.  II.] 


MEASUREMENT  OF  ANGLES. 


183 


GEOMETRICALLY. 


With  tlie  three  given  sides  construct 
the  triangle  ABC.  Then,  at  A lay  off 
the  angle  BAD  — 22°  30',  and  at  B the 
angle  ABD  = 33°  45',  and  note  D,  the 
point  at  which  the  two  lines  intersect. 

Through  the  points  A,  D,  and  B,  de- 
scribe the  circumference  of  a circle,  and 
through  C and  D draw  the  line  CDP ; 
the  point  P in  which  it  intersects  the 
circumference,  will  be  the  position  of  the  station. 

By  observing  the  equal  angles  in  the  figure,  the  trigono- 
metrical solution  is  not  difficult.  We  find, 


fPA=  710.198  yards. 

Ans.  \ PC  = 1042.524  “ 

| PB  = 934.289  “ 

Note. — This  problem  is  much  used  in  maritime  surveying, 
for  the  purpose  of  locating  buoys  and  sounding-boats.  The 
trigonometrical  solution  is  somewhat  tedious,  but  the  geomet- 
rical solution  is  very  easy,  as  shown  above. 


184 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


SECTION  III. 

RANGING  OUT  LINES,  ETC. 

204.  To  range  out  a line  with  the  transit,  place  the  instru- 
ment, carefully  adjusted,  over  the  first  station ; direct  the 
telescope  to  a distant  and  well-defined  point  in  the  desired  line, 
and  clamp  both  the  vernier  plate  and  the  horizontal  limb. 
The  line  of  sight  of  the  telescope  is  then  in  the  vertical  plane 
of  the  given  line,  so  that  points  on  the  surface  bisected  by  the 
intersection  of  the  cross-wires  will  be  in  the  required  line;  let 
an  assistant,  directed  into  the  line  by  the  observer  at  the  transit, 
fix  ranging-rods,  or  stakes  conspicuously  marked,  as  far  as  the 
power  of  the  telescope  extends.  Remove  the  transit  to  the 
third  or  fourth  stake  from  the  last  set,  and  place  it  precisely 
over  that  position  by  plumb-bob,  and  adjust  it  for  observation  ; 
the  telescope  is  ranged  in  the  line  by  sighting  backwards  and 
forwards  to  the  stakes  already  set.  The  line  is  then  continued 
as  before. 

If  great  accuracy  be  required,  each  operation  must  be 
repeated  with  telescope  reversed,  as  only  in  this  way  can  error 
in  adjustment  of  cross-wires  be  eliminated.  If  the  sighting 
with  reversed  telescope  does  not  agree  with  the  former  sighting, 
ake  a point  midway  between  the  two  points  sighted,  as  a point 
in  the  required  line. 

205.  A line  may  be  traced  in  forests  or  plantings,  in  which 
there  are  no  general  surface  obstructions,  by  the  aid  of  auxiliary 
parallel  lines. 

In  the  illustration,  Fig.  103,  an , bb , cc , are  the  parallel  lines, 
and  an , cx,  the  auxiliary  lines.  AB  is  a line  in  which  b is  a given 
point.  The  distances  ab , be,  in  this  lino  should  be  measured. 


SEC.  III.] 


BANGING  OUT  LINES. 


185 


and  also  the  angle  which  bb  makes  with  AB.  The  line  bb  and 
the  auxiliary  lines  should  be  traced  on  this  angle,  until  the 
trace  of  one  of  the  parallel  lines  be  obstructed  by  a tree. 


such  as  bb  at  (1).  Immediately  on  passing  the  obstruction 
on  one  of  the  auxiliary  lines,  a line  should  be  traced  on'  the 
measured  angle  or  its  supplement,  as  may  be  required,  and 
traced  to  intersect  the  other  auxiliary  line.  The  angle  made  by 
these  lines  should  be  measured  at  the  point  of  intersection  to 
verify  the  trace  of  the  intersecting  lines.  From  these  angular 
points  in  the  auxiliary  lines,  distances  to  the  point  (1),  equal  to 
ab  and  cb , respectively,  should  be  measured  in  the  transverse 
line,  and  found  to  meet,  but  not  overlie,  one  another.  Then 
will  the  point  of  meeting  in  the  transverse  line  be  a forward 
point  in  the  line  bb.  At  a suitable  distance  forward  from  which 
the  point  (1)  may  be  observed,  a like  determination  of  another 
point  in  bb  should  be  made.  The  trace  of  the  line  bb  should  be 
taken  upon  these  points  and  continued  in  connection  with  the 
auxiliary  lines  until  the  trace  of  one  of  the  lines  be  obstructed, 
such  as  the  line  an  at  (2).  The  trace  of  the  obstructed  line 
should  be  taken  up  by  measurements  in  the  transverse  line 


186 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV. 


BXAX,  and  in  a forward  parallel  line,  and  the  traces  continued  as 
described  above.  In  like  manner  the  obstructions  at  (3),  (4), 
&c.,  may  be  passed,  and  the  trace  of  the  line  continued  for 
considerable  distances  with  sufficient  accuracy  for  most  practical 
purposes.  The  continued  parallelism  of  the  lines  at  the  meas- 
ured distance  apart  will  be  a sufficient  verification  (Smith’s 
Treatise  on  Land  Surveying.) 


206.  To  measure  distances  by  means  of  the  transit. — The 

cross-wire  ring  in  the  telescope  of  the  transit  is  often  fitted  with 
an  arrangement  called  “ Stadia,”  or  “Micrometer.” 


The  Stadia,  or  Mi- 
crometer, is  a compound 
cross- wire  ring  or  dia- 
phragm, shown  in  Fig- 
ures 104  and  105,  having 
three  horizontal  wires,  of 
which  the  middle  one  is 
cemented  to  the  ring  as 
usual,  while  the  others, 
bb  and  cc,  are  fastened  to 

small  slides,  held  apart  by  a slender  brass  spring  hoop,  and 
actuated  by  independent  screws,  dd,  by  which  the  distance 
between  the  two  movable  wires  can  be  adjusted  to  include  a 
given  space  upon  a rod,  held  perpendicular  to  the  optical  axis 
in  front  of  the  object-glass,  at  a distance  from  it  equal  to  its 
principal  focal  distance. 

When  the  wires  are  thus  adjusted  to  include  a certain  space, 
as  two  feet  for  instance,  upon  a rod  placed  at  a distance  of 
100  feet  from  the  specified  point  on  the  optical  axis,  it  is  found 
that  they  will  cover  one  foot  at  half  that  distance,  or  four  feet 
at  a distance  of  200  feet;  thus  the  distance  is  proportional  to 
the  space  intercepted  upon  the  rod.  By  adding  to  the  distance 


SEC.  III.] 


RANGING  OUT  LINES. 


18? 


thus  obtained,  the  principal  focal  distance  of  the  object-glass, 
plus  the  distance  of  the  object-gJass  from  the  middle  of  the 
horizontal  axis,  the  distance  of  the  rod  from  the  station  can  be 
ascertained  without  the  use  of  a chain. 

The  focal  distance  of  the  object-glass  can  be  readily  obtained 
by  sighting  some  very  distant  object,  being  careful  to  correct 
instrumental  parallax  (Art.  185),  and  then  measuring  the  dis- 
tance from  the  object-glass  to  the  capstan  screws  of  the  cross- 
wire ring,  which  call  a ; now  sight  some  object,  distant  about 
100  feet,  and  measure  the  distance  from  the  object-glass  to  the 
horizontal  axis,  which  call  b ; the  sum  of  these,  a + b,  will  be  a 
constant,  sufficiently  exact,  to  be  added  to  all  distances  obtained 
by  readings  on  the  rod. 

The  spaces  upon  the  rod  used  should  be  equal  to  that  which 
the  instrument  intercepts  at  100  feet  from  the  point  in  front  of 
the  object-glass,  and  should  be  numbered  from  the  bottom  up ; 
each  space  should  be  subdivided  into  hundredths.  The  rod 
should  have  two  movable  targets,  like  those  used  upon  leveling 
rods,  and  should  also  be  furnished  with  an  attached  plumb,  or 
level,  to  insure  its  vertical  position.  A distinct  mark  should  be 
made  upon  it  at  the  ordinary  height  of  the  horizontal  axis  of 
the  instrument. 

In  using  the  micrometer,  sight  the  middle  horizontal  hair  to 
the  “ height  of  instrument”  mark,  and  then  direct  the  targets  to 
be  moved  successively  till  they  coincide  with  the  micrometer 
wires,  the  rod  being  kept  vertical.  If  the  telescope  has  been 
level  during  this  operation,  the  distance  given  by  the  rod  plus 
the  instrument  constant,  can  be  recorded  ; but  if  the  line  of 
sight  has  been  elevated  or  depressed,  then  from  the  distance 
given  by  the  rod,  including  the  instrument  constant,  must  be 
subtracted  the  product  of  this  distance  and  the  square  of  the 
sine  of  the  angle  of  deviation  from  the  horizontal. 

If  distances  greater  than  000  feet  are  to  be  measured,  the 


188 


ELEMENTS  OF  SURVEYING. 


BOOK  JV. 


unit  of  the  rod  must  be  less  than  ^ of  the  standard  distance, 
to  avoid  the  use  of  a rod  too  long  for  practical  management. 

If  the  distances  are  to  be  recorded  in  chains  and  links,  set 
the  stadia  staff  at  66  feet  in  order  to  obtain  its  unit,  and  then 
graduate  it  to  this  unit,  and  subdivide  to  hundredths. 

The  stadia  staff  should  be  spiked,  that  it  may  be  thrust  into 
the  ground  to  secure  steadiness. 

The  telescope  should  be  a good  one,  giving  a very  sharp, 
clear  definition  of  objects,  and  the  micrometer  wires  should  be  very 
fine  indeed  in  order  to  secure  close  readings. 

The  degree  of  accuracy  that  may  be  attained  is  shown  by  the 
following  table,  deduced  from  that  given  in  “Cours  de  Topo- 
graphic, par  A.  Lehagre,  1881,”  to  which  the  student  is  referred 
for  a very  full  description  of  the  method  : 

Focal  length  of  object-glass  in 

inches  6 8 10  12  14 

Distance  in  feet  which  may  be 

safely  measured  ....  600  800  1000  1200  1400 

Relative  error  in  horizontal  dis- 

^ance ¥0~0  ¥0'0  ToVo  TWO  T4F0 

With  a telescope  of  ten  inches  focal  length  or  over,  and 
within  the  above  limits,  the  stadia  measurements  are  as  reliable 
as  chain  measurements  on  fairly  level  ground,  and  are  much 
more  accurate  than  chain  measures  on  bad,  broken  country. 
As  the  use  of  the  micrometer  requires  care  and  consumes  time, 
it  is  not  recommended  for  short  distances,  except  on  bad  ground, 
swamps,  inaccessible  distances,  &c. 

207.  In  connection  with  the  chain  or  tape,  the  transit  is  used 
to  obtain  horizontal  distances  on  sloping  ground.  The  chaining 
is  made  on  the  surface  of  the  sloping  ground,  and  not  by 
elevating  the  chain  as  described  in  Art.  71 ; and  the  angle  of 


SEC.  III.] 


RANGING  OUT  LINES. 


189 


the  slope  is  taken  with  the  transit,  by  marking  on  a rod  the 
vertical  distance  from  the  horizontal  axis  of  the  telescope  to 
the  ground,  and  sighting  to  this  mark  on  the  rod  held  vertical 
at  the  end  of  the  line  measured ; the  horizontal  distance  is  equal 
to  the  measured  distance  multiplied  by  the  cosine  of  the  angle 
of  the  slope. 

208.  To  survey  a line,  such  as  a road,  boundary  of  an  estate, 
&c.,  measure  the  angle  of  deviation  which  each  line  makes  with 
the  preceding  line  prolonged,  or  measure  the  azimuths  which 
each  line  makes  with  the  first  line  taken  as  a meridian  (Art.  194), 
and  measure,  also,  the  length  of  each  line  and  offsets  to  prominent 
objects.  Care  must  be  taken  to  centre  the  instrument  exactly 
over  each  angular  point,  as  any  error  in  centreing  will  cause  an 
error  in  the  apparent  direction  of  the  object  sighted,  which  will 
be  the  greater  the  nearer  the  object  is  to  the  instrument. 

209.  To  survey  the  streets  of  a town  or  city,  place  the  transit 
at  the  intersection  of  two  or  more  of  the  principal  streets, 
through  which  the  longest  lines  of  sight  can  be  had  ; find  the 
angle  which  each  of  the  streets  diverging  from  this  point  makes 
with  the  principal  street,  and  find,  also,  the  angle  of  slope  of 
each  of  the  streets  at  this  point ; measure,  with  the  chain  or 
stadia-rod,  or  both  as  checks  one  upon  the  other,  the  lengths 
of  the  lines  of  sight,  and  take  offsets  to  the  corners  of  all  streets, 
to  public  buildings  and  prominent  objects  ; remove  the  transit 
to  the  next  street  and  take  the  angles,  angle  of  slope,  measure- 
ments, and  offsets  as  before,  and  so  continue  till  the  survey  is 
complete. 


190 


ELEMENTS  OF  SURVEYING. 


[ROOK  IV. 


SECTION  IV. 


FARM  SURVEYING  BY  TRANSIT. 


210.  The  figure  and  area  of  any  piece  of  ground  may  be 
found  by  beginning  at  any  one  of  the  angular  points  and  going 
entirely  around  the  boundary,  measuring  the  length  of  the  sides 
by  chain  or  stadia-rod,  and  the  angle  which  each  side  makes  with 
the  preceding  side  prolonged,  called  Angle  of  Deviation,  or  the 
azimuths  of  the  several  sides  with  a given  first  side  as  meridian. 
Let  the  farm  to  be  surveyed  be  the  one  given  in  Article  123. 
Let  the  side  AB  be  taken  as  the  meridian  of  the  survey. 
Measure  with  the  transit  the  azimuths  of  the  several  successive 
sides  with  A B,  as  directed  in  Art.  194,  and  enter  them  in  the 
notes  at  the  left  of  the  station  mark.  In  the  following  illustration 
both  the  azimuth  and  the  angle  of  deviation  have  been  entered, 
though  the  surveyor  would  use  but  one,  together  with  the 
bearings,  which  should  always  be  entered  as  a check : 


Azimuths.  Angles  of  Deviation. 


99°  30'  123°  30'  y D 


336°  00'  100°  55’  ''I  0 

76°  55'  76°  55'  Y B 


A 


Bearings. 

N 30°  35'  E 


4 

22.89 

A 

1.40 


(N  87°  5'  E) 

S 87°  5'  W 
(S  7°  50'  W) 


A 

31.95 


N8°E 
(S  68°  55'  E) 


A 


N G8°  55'  W 


From  the  azimuths  determine  the  bearings  of  the  several  sides 
with  AB , as  directed  in  Art.  195,  and  let  them  be  as  noted 
on  the  following  page ; complete  the  table  and  determine  the  area 
as  in  a compass  survey  : 


SEC.  IV.] 


FARM  SURVEYING  BY  TRANSIT. 


191 


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192 


elements  of  surveying. 


[book  iv. 


211.  If  the  surveyor  does  not  record  compass-bearings,  as 
advised  to  do  in  last  article,  the  notes  may  be  kept  in  precisely 
the  same  manner  as  the  field  notes  of  the  compass  survey,  as 
shown  in  Article  123— substituting  azimuths  for  the  hearings 
of  the  several  courses,  and  angles  instead  of  hearings  for  the 
prominent  objects  sighted  to. 

212.  If  the  angles  of  deviation  had  been  measured 
instead  of  the  azimuths,  the  method  of  finding  the  area  would 
have  been  substantially  the  same.  The  azimuths  would  have 
been  determined  from  the  angles  measured,  and  the  rest  of  the 
computation  would  have  remained  the  same.  The  azimuth  of 
any  course  with  a course  taken  as  meridian  may  be  found  from 
the  angles  of  deviation,  thus:  The  azimuth  of  the  course  taken 
as  meridian  is  zero;  the  azimuth  of  the  second  course  is  equal 
to  its  angle  of  deviation  ; the  azimuth  of  any  succeeding  course 
is  equal  to  the  azimuth  of  the  preceding  course  increased  by  the 
angle  of  deviation  of  the  course  itself,  as  a simple  diagram 
will  show.  If  the  interior  angles  had  been  measured,  the 
angles  of  deviation  would  have  been  obtained  by  subtracting 
each  interior  angle  from  180°  and  the  method  would  have  been  • 


The  other 
columns  as 
before. 


Stations. 

Interior  Angles. 

Angles  of  Deviation. 

Azimuth  with  AB. 

A 

98°  55' 

81°  05' 

0 

B 

103°  05' 

76°  55' 

76°  55' 

0 

280°  55' 

— 100°  55' 

(-24°)  336° 

D 

56°  30' 

123°  30' 

99°  30' 

B 

168°  30' 

11°  30' 

111° 

F 

135° 

45° 

156° 

G 

151°  35' 

ib 

o 

OO 

C2 

184°  25' 

II 

180°  30' 

— 6°  30' 

177°  55' 

I 

13G° 

44° 

221°  55' 

K 

123° 

Cl 

1 o 

278°  55' 

SEC.  IV.] 


FARM  SURVEYING  BY  TRANSIT. 


193 


213.  When  an  angle  of  deviation  lies  within  the  boundary  of 
the  survey,  as  at  G,  (see  page  87),  it  must  be  called  negative ; 
when  it  lies  without  the  boundary,  it  is  an  exterior  angle  of  the 
polygon  and  is  positive.  The  azimuth  obtained  by  the  rule  may 
be  negative  (as  it  is  in  the  case  of  the  course  CD) ; but  as  the 
azimuth  must  be  positive,  360°  must  be  added  to  the  negative 
result  obtained  to  get  the  true  azimuth. 

The  algebraic  sum  of  the  angles  of  deviation  should  be  equal 
to  or  differ  but  little  from  360°,  which  fact  serves  to  check  the 
correctness  of  the  angles  recorded. 

214.  Where  all  the  corners  of  a field  may  be  seen  from  one 
of  them,  the  area  may  be  determined  as  follows : 

Place  the  transit  over  the  point  A , 

Fig.  106,  from  which  the  other  corners 
may  be  seen,  and  measure  the  angles  BA  C \ 

CAD,  and  DAE.  Measure  the  lengths  of 
the  diagonals  AC  and  AD,  and  of  the 
sides  AB  and  AE ; then  the  areas  of  the 
triangles  ABC,  CAD,  and  DAE,  may 
be  separately  found  from  the  principle 
that  the  area  of  a triangle  is  equal  to  half  the  product  of  two 
sides  and  the  sine  of  their  included  angle  ; the  sum  of  the  areas 
of  the  triangles  is  equal  to  the  area  of  the  field. 

215.  If  a description  of  the  property  is  also  required,  the 
bearing  of  some  one  of  the  lines  must  be  taken,  and  from  this 
and  the  recorded  angles  and  distances,  the  “Bearings  and 
Courses”  may  be  computed. 

216.  Where  all  the  corners  of  the  field  may  be  seen  from  a 
point  within,  as  A,  Fig.  107.  Place  the  transit  over  the  point  A ; 
measure  the  angles  BAG,  CAD,  &c.,  at  A,  and  the  length 


B 


104 


ELEMENTS  OF  SURVEYING. 


[BOOK  IV 


of  the  lines  AB , AC,  &c.,  from 
A to  the  several  corners  of  the 
field  ; find  the  areas  of  the  tri- 
angles BAC,  CAD,  &c.,  as  in 
the  last  article,  and  add  them 
together.  If  the  boundary  is 
irregular,  as  represented  in  the 
figure,  measure  offsets,  calculate 
the  contents  of  these  smaller 
portions  separately,  and  add  or 
subtract  them  as  may  be  nece 
the  tract. 


C 


ry,  to  find  the  true  area  of 


217. — e x a m p l e s . 

1.  Required  the  contents  and  plot  of  a piece  of  land,  of 
which  the  following  are  the  field  notes  : 


Stations. 

Azimuths  with  AB. 

Distances. 

A 

0° 

15.8  ch. 

B 

o 

CO 

17.4 

C 

© 

CO 

o 

o 

32.56 

D 

154°  30' 

14.88 

E 

189° 

24.96 

F 

230° 

14. 

G 

279° 

32.8 

A 

360° 

2.  Required  the  contents  and  plot  of  a piece  of  land,  of  which 
the  following  are  the  field  notes  : 


SEC.  IV  ] 


FARM  SURVEYING  BY  TRANSIT. 


195 


Station?. 

Interior  Angles. 

Distances. 

A 

92°  30' 

31.80  ch. 

B 

94°  30' 

2.08 

C 

155°  15' 

2.21 

D 

179°  30' 

35.35 

E 

94°  15' 

21.10 

F 

104° 

31.30 

3.  Required  the  area  of  a piece  of  land,  of  which  the  follow- 
ing measurements  were  made : 


AB 

AC 

AD 

AE 

AF 

angle  BAC 
CAD 
DAE 
EAF 


= 20  chains 
= 22.57 
= 28.64 
= 40.80 
= 30.95 

= 37° 

= 46°  45' 

= 42°  15' 

= 26° 


Offsets  from  the  line  AB  were  taken  as  follows: 


At  3.00  ch. 
“ 6.50  “ 

“ 9.50  “ 

“ 13.00  “ 

“ 16.00  “ 

« 20.00  “ 


R offset  2.50  ch. 

“ 0.00  “ 

L “ 1.60  “ 

L “ 2.00  “ 

L “ 1.40  “ 

“ 0.00  « 


BOOK  V. 

LAYING  OUT  AND  DIVIDING  LAND. 


SECTION  I. 

OF  DIVIDING  LAND. 

218.  The  surveyor  is  often  required  to  lay  off  a given 
quantity  of  land,  in  such  a way  that  its  bounding  lines  shall 
form  a particular  figure,  viz.,  a square,  a rectangle,  a triangle, 
&c.  He  is  also  often  called  upon  to  divide  given  pieces  of 
land  into  parts  containing  given  areas,  or,  into  areas  bearing 
certain  relations  to  each  other. 

The  manner  of  making  such  divisions  must  always  depend 
on  a skilful  and  judicious  application  of  the  principles  of 
geometry  and  trigonometry  to  the  particular  case. 

For  example,  if  it  were  required  to  lay  out  an  acre  of 
ground,  in  a square  form,  it  would  be  necessary  to  find,  by 
calculation,  the  side  of  such  a square,  and  then  trace,  on  the 
ground,  a figure  bounded  by  four  equal  sides  at  right  angles 
to  each  other. 

219.  To  lay  out  a given  quantity  of  land  in  a square  form. 

Rule. — Reduce  the  given  area  to  square  chains  or 
square  rods ; then  extract  the  square  root , and  the  result 
will  be  the  side  of  the  required  square.  This  square 
being  described  on  the  ground,  will  be  the  figure  required. 


SEC.  I.] 


OF  DIVIDING  LAND. 


197 


220.  To  lay  out  a given  quantity  of  land  in  a rectangular 
form,  when  one  of  the  sides  of  the  rectangle  is  given. 

Rule. — Divide  the  given  area , reduced  to  square  chains 
or  square  rods,  by  tlve  given  side  of  the  required  rectangle, 
and  the  quotient  will  be  the  other  side . Then,  trace  the 
rectangle  on  the  ground. 

221.  To  run  a line  from  the  vertex  of  a triangular  field  which 

shall  divide  it  into  two  parts,  having  to  each  other  the 

ratio  of  m to  n. 

Let  ABC  be  any  triangular  field. 

Divide  the  side  BC  into  two  parts, 
such  that  (Geom.,  Bk.  IY,  Prob.  I) 

BD  : DC  : : m : n; 
and  draw  the  line  AD ; 
then  will  ABD  : DAC  ::  m : n. 

For,  the  two  triangles  ABD , ADC , having  the  same  alti- 
tude, are  to  each  other  as  their  bases  (Geom.,  Bk.  IY,  P.  6,  C.)  ; 
hence,  the  triangle  is  divided  into  parts  having  the  ratio  of 
m to  n. 

222.  To  run  a line  parallel  to  one  side  of  a triangular  field, 

that  shall  form  with  the  parts  of  the  two  other  sides  a 

771 

triangle  equal  to  the  — part  of  the  field. 

Let  CBA  represent  a triangu- 
lar field,  and  CA  the  side  parallel 
to  which  the  dividing  line  is  to  be 
drawn. 

On  the  side  BC,  take  BE  equal 


Fig.  110. 


198 


ELEMENTS  OF  SURVEYING. 


[BOOK  V. 


to  BC  \ and  on  the  side  BA,  take  BF  equal  to  BA  i/—  ; 
n irn 

the  line  EF  is  the  line  required;  for,  since  it  divides  the  sides 

BC  and  BA  proportionally,  it  is  parallel  to  the  side  CA  (Geom., 

Bk.  IV,  P.  XVI)  ; and  from  the  similar  triangles,  we  have 

(Geom.,  Bk.  IV,  P.  XXV), 

BEF  : BCA  : : BE2  : BC \ 

or  BEF  : BCA  : : m : n ; 

hence,  BEF  = - BOA. 


Example. — Let  it  be  required  to  divide  the 
triangular  field  CAB,  in  which  AC—  9 ch., 
AB  = 11  ch.,  and  CB  = 7 ch.,  into  two  such 
parts  that  ADE  shall  be  one-fourth  of  the 
whole  field. 

In  this  case,  we  have, 


A 


m — 1,  n = 4, 


1 

2 ’ 


hence,  AE  = 4 ch.  50  1.,  and  AD  = 5 ch.  50  1. 


223.  To  run  a line  from  a given  point  in  the  boundary  of  a 
piece  of  land,  so  as  to  cut  off,  on  either  side  of  the 
line,  a given  portion  of  the  field. 

Make  a complete  survey  of  the  field,  by  the  rules  already 
given.  Let  us  take,  as  an  example,  the  field  whose  area  is 
computed  in  Ex.  1,  Art.  140.  That  field  contains  105  A.  2 R. 
33  P.,  and  Fig.  1 12  is  a plot  of  it. 

Let  it  now  be  required  to  run  a line  from  station  A in 
such  a manner  as  to  cut  off,  on  the  left,  any  part  of  the  field ; 
sav,  2G  A.  2 R.  31  P. 


SEC.  I.] 


OF  DIVIDING  LAND. 


199 


It  is  seen,  by  exam- 
ining the  field,  that  the 
division  line  will  proba- 
bly terminate  on  the 
course  CD.  Therefore, 
draw  a line  from  A to 
C,  which  we  will  call 
the  first  closing  line.  fig.  112. 

The  bearings  and 

lengths  of  the  courses  AB,  BC,  are  always  known,  and  in  the 
present  example  are  found  in  the  table  Art.  140,  Ex.  1 ; hence, 
the  bearing  and  distance  from  C to  A can  be  calculated  by  Art. 
142  ; they  are,  in  this  example, 

Bearing,  S 9°  28'  E;  Course,  23.22  ch. 

Having  calculated  the  bearing  and  length  of  the  closing 
line,  find,  by  the  general  method,  the  area  which  it  cuts  off; 
that  area,  in  the  present  case,  is  14  A.  0 R.  26  P. 

It  is  now  evident  that  the  division  line  must  fall  on  the 
right  of  the  closing  line  AC,  and  must  cut  off  an  area  ACH , 
equal  to  the  difference  between  that  already  cut  off,  and  the 
given  area;  that  is,  an  area  equal  to  12  A.  2 R.  5 P. 

Since  the  bearing  of  the  next  course  CD,  and  the  bearing 
of  the  closing  line  AC  are  both  known,  the  angle  ACD  which 
they  form  with  each  other,  can  be  calculated,  and  is  in  this 
example,  79°  32'.  Hence,  knowing  the  hypothenuse  AC,  and 
the  angle  ACG  at  the  base,  the  length  AG,  the  perpen- 
dicular let  fall  on  the  course  CD  can  be  found,  and  is  22.82 
chains. 

The  base  of  a triangle  is  equal  to  its  area  divided  by  half 
the  altitude.  Therefore,  if  the  area  12  A.  2 R.  5 P.  be  re- 
duced to  square  jhains,  and  divided  by  11.41  chains,  which 


200 


ELEMENTS  OF  SURVEYING. 


[HOOK  V. 


is  half  the  perpendicular  AG,  the  quotient,  which  is  10.95 
chains,  will  be  the  base  CH.  Hence,  if  we  lay  off  from  C,  on 
CD,  a distance  CH,  equal  to  10.95  chains,  and  then  run  the 
line  AH,  it  will  cut  off,  from  the  land,  the  required  area,  viz., 

26  A.  2 R.  31  P. 


Note  1. — If  the  part  cut  off  by  the  first  closing  line 
should  exceed  the  given  area,  the  division  line  will  fall  on  the 
left  of  AC. 

Note  2.- — If  the  difference  between  the  given  area  and  the 
first  area  cut  off,  divided  by  half  the  perpendicular  AG,  gives 
a quotient  larger  that  the  course  CD  ; then,  draw  a line  from 
A to  D,  and  consider  it  as  the  first  closing  line,  and  let  fall  a 
perpendicular  on  DE. 

Note  3. — When  the  point  from  which  the  division  line  is  to 
be  drawn  falls  between  the  extremities  of  a course,  divide  the 
course  into  two  parts,  at  this  point.  Then  consider  one  of  the 
parts  as  an  entire  course,  and  the  other  as  forming  a new  course, 
having  the  same  bearing.  The  manner  of  making  the  calcula- 
tion will  then  be  the  same  as  before. 


224.  To  cut  off  from  a field,  a given  area,  by  a line  running 
in  a given  direction. 

In  this  case,  as  in  the  previous  one,  a complete  and  correct 
survey  is  first  necessary.  Then,  when  the  whole  area  is  known, 
the  position  of  the  line  may  be  approximately  determined  by  the 
inspection  of  a correct  map  of  the  whole. 

Wo  will  take,  for  illustration,  Example  4,  Art.  140,  of  which 
Fig.  113  is  the  plot. 


SEC.  T.] 


201 


OF  DIVIDING  LAND. 


V 


Let  it  be  required  to  cut  off  from  this  area,  £0  acres,  by  a 

line  whose  bearing  shall  be  S or  N 00°  W. 

SZ, 

We  will  make  a trial  of  a line  starting  at  25  chains  from 
station  6,  on  the  6th  course.  We  will  call  this  station  A,  and 
the  trial  line  AB. 


In  order  to  determine  if  the  area  cut  off  is  equal  to  the 
required  area,  we  must  first  determine  the  length  of  AB  and 
of  B 5.  These  cannot  be  determined  by  the  method  of  sup- 
plying lost  notes. 

We  must  first  calculate  the  length  of  a line,  starting  at  the 
proposed  point,  and  running  to  the  station  nearest  to  the  other 
extremity  of  the  closing  line.  In  this  example,  from  A to  5. 
This  is  easily  found  to  be  36.406  chains,  and  its  bearing 
N 81°  13'  E. 


F 


ELEMENTS  OF  SURVEYING. 


[LOOK  V. 


202 


Now,  in  the  triangle  AB  5 we  have  one  side  and  the  angles, 
to  find  the  remaining  parts.  AB  is  found  to  be  28.88  chains 
and  B 5 to  be  22.81  chains.  We  have  now  the  complete  field- 
notes  of  the  area  cut  off. 


A 

o 

© 

GO 

28.88  ch. 

B 

P3 

o 

QO 

CQ 

ft 

22.81 

5 

N 57°  W 

21.10 

G 

S 47°  W 

25.00 

The  area  is  found 
to  be  58.5029  acres. 


It  now  remains  to  move  this  line  northerly,  so  that  the  area 
contained  between  its  present  position  and  the  new  one  shall  be 
equal  to  8.5029  acres. 

Suppose  the  lines  A G and  B 5 be  prolonged  till  they  meet  at 
some  point,  as  V , Fig.  114. 

Calculate  A V and  BV,  also  the  area  ABV. 

A V is  found  to  be  92.19  chains  and  BV  88.18 
chains.  The  area  of  the  triangle  ABV , is  127.29 
acres.  Let  MN  represent  tbe  line  sought. 

Then,  we  have  two  similar  triangles,  with  all 
the  sides  of  the  one,  and  the  areas  of  each, 
known  ; for,  Fdfi^must  contain  8.5029  acres  less 
than  A VB.  Then,  AM  and  BN  are  easily 
determined. 

The  complete  notes  of  the  area  to  be  cut  off,  are 


Fig.  114. 


M 

S G0°  E 

27.89 

N 

N 28J°  E 

19.82 

5 

N 57°  W 

21.10 

G 

S 47°  W 

21.87 

SEC.  II.] 


PUBLIC  LAHDS. 


203 


Note. — Fields  are  so  variously  shaped  that  it  is  difficult  to 
give  rules  that  will  apply  to  all  cases.  It  is  by  practice  alone 
that  facility  is  obtained  in  that  branch  of  surveying  relating  to 
the  division  of  estates.  We  have  given  only  a few  examples 
that  may  serve  as  general  guides  in  the  application  of  the 
principles  of  Plane  Geometry  to  such  cases  as  may  arise. 


SECTION  II. 

PUBLIC  LANDS  OF  THE  UNITED  STATES. 

225.  Soon  after  the  organization  of  the  government,  several 
of  the  States  ceded  to  the  United  States  large  tracts  of  wild  land, 
and  these,  together  with  the  lands  since  acquired  by  treaty  and 
purchase,  constitute  the  public  lands,  or  public  domain.  These 
lands  were  at  first  parceled  out  without  reference  to  any  general 
plan,  in  consequence  of  which  the  titles  often  conflicted  with 
each  other,  and  in  many  cases,  several  grants  covered  the  same 
area.  Through  many  years  of  labor  and  experiment,  the  present 
admirable  system  has  been  wrought  out. 

226.  This  system  is  briefly  that  the  territory  to  be  surveyed 
shall  be  divided  by  true  north  and  south,  and  east  and  west  lines, 
into  tracts,  each  of  six  miles  square  and  containing  as  near  as 
may  be  23,040  acres,  called  Townships ; each  township  into 
thirty-six  tracts,  each  of  one  mile  square  and  containing  as  near 
as  may  be  640  acres,  called  Sections ; and  each  section  into 
halves,  quarters,  and  smaller  portions,  as  may  be  deemed 
expedient. 

227.  In  the  survey,  all  primary  lines  north  and  south  are 


ELEMENTS  OF  SURVEYING. 


204 


[book  V. 


run  on  meridians  of  longitude,  and  all  primary  lines  east  and 
west  are  laid  out  on  perpendiculars  to  the  meridians. 

It  is  of  the  first  importance,  then,  that  meridians  of  longitude 
should  be  accurately  determined,  and  that  lines  perpendicular  to 
them  should  be  determined  with  equal  precision. 

The  ordinary  Surveyor’s  Compass  is,  for  several  reasons,  not 
sufficiently  accurate  for  the  work  of  running  out  the  standard 
and  township  lines,  and  such  lines  must  be  run  by  Burt’s  Solar 
Compass,  “or  other  instrument  of  equal  utility.” 

228.  The  Solar  Compass  for  determining  a true  meridian 
was  invented  by  William  A.  Burt,  of  Michigan,  and  patented  by 
him  in  1836.  It  has  been  improved,  by  him  and  others,  from 
time  to  time  since,  and  in  its  present  state  is  represented  and 
described  in  Appendix  A. 

229.  In  commencing  the  division  of  the  public  lands  in 
unsurveyed  territory,  an  initial  point  is  selected,  with  reference 
to  its  convenience  in  making  the  survey,  perpetuated  by  a 
substantial  monument  suitably  marked,  and  its  true  position  in 
latitude  and  longitude  determined. 

230.  From  the  initial  point  the  principal  base  line  is  run 
out  due  east  and  west  with  the  solar  compass,  and  permanently 
marked  at  each  40  chains,  or  half  mile,  with  a quarter-section 
corner,  and  at  each  80  chains,  or  mile,  with  a section  corner. 

231.  The  principal  meridian  is  then  run  out  due  north  and 
south  from  the  initial  point,  and  also  marked  with  monuments 
at  intervals,  like  the  base  line. 

232.  As  meridians  of  longitude  converge  toward  the  poles, 
the  distance  between  two  such  meridians  decreases  as  the 
surveyor  goes  north.  To  counteract  the  error  that  would  other- 
wise result  from  the  convecgency  of  meridians,  and  also  to  arrest 


SEC.  II.] 


PUBLIC  LANDS. 


205 


error  arising  from  inaccuracies  in  measurements  on  meridian 
lines,  standard  parallels , or  correction  lines,  are  run  east  and 
west  from  the  principal  meridian  and  at  stated  intervals.  On  the 
north  of  the  principal  base-line,  about  latitude  35°  north,  these 
standard  parallels  are,  in  general,  run  at  distances  of  every  four 
townships,  or  twenty-four  miles,  and  south  of  the  principal  base, 
at  distances  of  every  five  townships,  or  thirty  miles.  Each  of 
these  standards  is  run  out  and  marked  in  the  same  way  as  the 
principal  base,  and  forms  the  base  for  laying  out  the  townships 
north  to  the  next  standard  parallel.  The  standards  are  num- 
bered according  to  their  position  with  respect  to  the  principal 
base-line,  as  1st  Standard  Parallel  South,  2d  Standard  Parallel 
South,  1st  Standard  Parallel  North,  &c. 

233.  The  principal  meridian,  the  base-line,  and  the  standard 
parallels  having  been  first  run,  measured,  and  marked,  and  the 
corner  boundaries  thereon  established,  the  exterior  lines  of 
townships  are  then  run,  measured,  and  marked. 

The  townships,  consisting  of  a series  of  townships  lying  along 
a parallel,  are  numbered  north  or  south  of  the  principal  base ; 
the  first  series  north  of  the  base  being  Township  1 North,  the 
second  series  north  being  Township  2 North,  &c. ; and  the  first 
series  south  being  Township  1 South,  &c.;  these  are  designated 
T.  1 N.,  T.  2 N.,  T.  1 S.,  &c. 

The  ranges,  consisting  of  tiers  of  townships,  are  numbered  from 
the  principal  meridian  both  ways;  the  first  tier  west  of  the 
meridian  being  Range  1 West,  the  first  tier  east  being  Range  1 
East,  &c.;  designated  R.  1 W.,  R.  1 E.,  &c. 

234.  The  accompanying  map,  from  the  U.  S.  General  Land 
Office,  representing  a considerable  portion  of  the  State  of 
Arkansas,  will  serve  for  illustration. 

The  principal  meridian  in  this  survey  is  called  the  5th 


20G 


ELEMENTS  OF  SURVEYING. 


[BOOK  V 


SEC.  II.] 


PUBLIC  LANDS. 


207 


meridian,  and  passes  through  the  point  of  junction  of  the  White 
river  with  the  Mississippi.  The  principal  base-line,  running  east 
and  west,  intersects  this  meridian  a little  to  the  east  of  White 
river;  and  from  the  meridian  and  base-line,  reckoned  from 
this  point  of  intersection,  all  the  ranges  of  townships  are 
laid  off. 

For  example,  1 North,  will  apply  to  all  the  townships  lying  in 
the  first  row  north  of  the  base-line  ; 1 South,  will  apply  to  all  the 
townships  in  the  first  row  south  of  the  base  line.  Range  1 East,  will 
apply  to  all  the  townships  lying  in  the  first  row,  east  of  the  5th 
meridian ; and  Range  1 West,  will  apply  to  all  lying  in  the  first  row 
to  the  west  of  it.  The  small  figures  designate  the  rows  of  town- 
ships, reckoned  north  and  south  from  the  base-line,  and  the  ranges 
reckoned  east  and  west  from  the  5th  meridian.  Thus,  Township 
1 North,  Range  4 West,  has  its  exact  place  designated,  and  may 
be  immediately  located. 

235.  The  diagram  here  given  (Fig.  116),  from  the  “ Instruc- 
tions to  the  Surveyors-General  of  Public  Lands  of  the  United 
States,”  represents  the  required  method  of  running  out  township 
lines. 

In  the  diagram,  the  upright  figures  (made  thus,  1,  2,  3) 
commencing  near  the  Principal  Meridian  and  Base  Line  with 
No.  1,  indicate  the  perambulations  of  the  Surveyor  in  running 
the  Townships  and  Correction  lines. 

The  Correction  or  Standard  lines  north  of  the  base  are  every 
four  townships,  and  south  of  the  base  every  five  townships. 

The  excess  or  deficiency  of  measurement  on  northern. and 
southern  boundaries  is  thrown  on  the  westernmost  half-mile. 

The  measurements  between  meridian  lines  will,  of  course, 
always  vary  according  to  the  latitude  of  the  survey,  besides 
being  liable  to  be  rendered  inexact  where  the  country  is  very 
hilly  or  broken.  The  convergency  of  the  range  lines  as  shown 


208 


ELEMENTS  OF  SURVEYING, 


[BOOK  V. 


by  the  measurements  on  this  diagram,  is  according  to  calculation, 
as  it  exists  between  the  parallels  of  46°  and  47°  North  Latitude. 


EXTERIORS 


Diagram. 

OR  I,  TOWNSHIP  LTNES 


Krsi 

80, 80  I 80  I 80 , 80,80 


correction  or 

00 ,80  I 80  I 80 , 80 ,80 


Standard  Line  NorGi 

80  80  I 80  I 80  I 80 ,80  80  80  80 , 80 


S2I 

17.1& 


80' 80  1 80  1 80  80 
11 


20 


m. 


80  1 80 1 80 ' 80  80 


80 1 80  80' 00  80 

11 


T.4.LL. 


17J& 


rm 


oo 1 no  1 no 


80  80 
22 


80 1 80 1 80 ‘80 '80 

8 


§18 


17 


T&fi \7/ 


80 1 80 1 80  80 


80  80  80  80 '80 
8 


M20 


80 ‘ 80 1 80 


80  80 


t.  s.jsr.  § 
§ 

6« 


o fi 


18 


80  1 80  1 80  1 80  80 

5 


80  1 80  1 80  1 80  '80 
5 


§17 


T.2  JST.  § 


J 

M 


M 


h8fl 


I 

4~ 


80  80  80  80  1 80 

2 


'80  80  1 80 


RJTW  i 

80  | 80  , 80 , 80  > 80 


TJ.N  g 
JUW  g 

80  ,80  i 80 , 80  ,80 


RLE  i 
oo  i no i no i no i no 


80 ' 30  1 80 


RILE 

80  , 80 , 30 


12 

80  ,80 


BftSO 


Lille 


Fia.  116. 

236.  The  “Instructions  to  the  Surveyors-General  ” for  run- 
ning, measuring,  and  marking  the  exterior  lines  of  tpwnships, 
are  as  follows: 


SEC.  II.] 


PUBLIC  LANDS. 


209 


Townships  situated  North  of  the  Base  Line  and  West  of  the 
Principal  Meridian. 

Commence  at  No.  1 (see  figures  on  the  diagram,  Fig.  116), 
being  the  southwest  corner  of  T.  1 N. — R.  1 W.,  as  established  on 
the  base  line ; thence  north  on  a true  meridian  line,  four  hun- 
dred and  eighty  chains  (6  miles),  establishing  the  section  and 
quarter-section  corners*  thereon,  as  per  instructions,  to  No.  2, 
whereat  establish  the  corners  of  Tps.  1 and  2 N. — Rs.  1 and  2 W.; 
thence  east,  on  a random  or  trial  line,  setting  temporary  section 
and  quarter-section  stakes  to  No.  3,  where  measure  and  note  the 
distance  at  which  the  line  intersects  the  eastern  boundary,  north 
or  south  of  the  true  or  established  corner.  Run  and  measure 
westward,  on  the  true  line  (taking  care  to  note  all  the  land  and 
water  crossings,  &c.,  as  per  instructions),  to  No.  4,  which  is 
identical  with  No.  2,  establishing  the  section  and  quarter-section 
permanent  corners  on  said  line.  Thence  proceed  in  a similar 
manner  from  No.  4 to  No.  5,  No.  5 to  No.  6,  No.  6 to  No.  7,  and 
so  on  to  No.  10,  the  southwest  corner  of  T.  4 N. — R.  1 W. 
Thence  north,  still  on  a true  meridian  line,  establishing  the 
mile  and  half  mile  corners,  until  reaching  the  Standard  Par- 
allel or  correction  line  ; throwing  the  excess  over,  or  deficiency 
under  four  hundred  and  eighty  chains , on  the  last  half  mile, 
according  to  law,  and  at  the  intersection  establishing  the 
“ Closing  Corner,”  the  distance  of  which  from  the  standard 
corner  must  be  measured  and  noted  as  required  by  the  instruc- 
tions. But  should  it  ever  so  happen  that  some  impassable  bar- 
rier will  have  prevented  or  delayed  the  extension  of  the  standard 
parallel  along  and  above  the  field  of  the  present  survey,  then 
the  depntv  will  plant,  in  place,  the  corner  for  the  township, 
subject  to  correction  thereafter,  should  such  parallel  be  ex- 
tended. 


210 


ELEMENTS  OF  SURVEYING. 


[BOOK  V. 


North  of  the  Base  Line  and  East  of  the  Principal  Meridian. 

Commence  at  No.  1,  being  the  southeast  corner  of  T.  IN. — 
E.  1 E.,  and  proceed  as  with  townships  situated  “north  and 
west,”  except  that  the  random  or  trial  lines  will  be  run  and 
measured  west,  and  the  true  lines  east,  throwing  the  excess  over 
or  deficiency  under  four  hundred  and  eighty  chains  on  the  ivest 
end  of  the  line,  as  required  by  law  ; wherefore  tht>  surveyor  will 
commence  his  measurement  with  the  length  of  the  deficient 
or  excessive  half-section  boundary  on  the  west  of  the  township, 
and  thus  the  remaining  measurements  will  all  be  even  miles  and 
half  miles. 

237.  In  running  random  township  exteriors,  if  such  random 
lines  fall  short  or  over-run  in  length,  or  intersect  the  eastern  or 
western  boundary,  as  the  case  may  be,  of  the  township,  at  more 
than  three  chains  and  fifty  links  north  or  south  of  the  true 
corner,  the  lines  must  be  retraced,  even  if  found  necessary  to 
re-measure  the  meridional  boundaries  of  the  township. 

238.  The  exterior  lines  of  townships  having  been  established 
and  duly  marked,  each  township  is  divided  into  36  squares, 
called  Sections,  by  meridians  one  mile  apart,  and  by  east  and 
west  lines  at  the  same  distance  from  each  other.  The  sections 
of  a township  are  numbered  from 
1 to  36,  beginning  at  the  north- 
east angle  and  proceeding  as  shown 
in  the  annexed  diagram  : 

To  describe  a section  accurately, 
we  say,  for  example,  section  num- 
ber 5,  in  township  number  4 north, 
in  range  3 west  of  a known 
meridian. 


6 

5 

4 

3 

2 

1- 

7 

8 

9 

10 

11 

12 

18 

17 

1G 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

3G 

239.  Tl  le  sections  are  divided 


Fio.  117. 


SEC.  II.] 


PUBLIC  LANDS. 


211 


into  half-sections,  quarter-sections,  and  even  into  eighths  of 
sections.  The  following  table  shows  the  contents  of  a township 
and  its  subdivisions  : 

1 township  = 36  sections  = 23040  acres. 

1 section  = 640  acres. 

\ section  = 320  acres. 

} section  = 160  acres. 

| section  = 80  acres. 

240.  As  any  excess  or  deficiency  of  measurement  is,  by  law, 
to  be  thrown  on  the  extreme  tier  of  sections  and  half-sections 
contiguous  to  the  north  and  west  boundaries  of  townships,  such 
sections  and  half-sections  are  sold  as  containing  only  the  quan- 
tity expressed  in  the  returns  and  plots,  respectively,  and  all 
others  as  containing  the  complete  legal  quantity. 

The  government  surveyors  are  rarely  required  to  subdivide  a 
section  into  quarters,  as  that  work  properly  belongs  to  the 
county  surveyors. 

Note. — The  student  is  referred,  for  more  detailed  information  on 
Government  Surveys,  to  : Instructions  to  the  Surveyors-General  of  Public 
Lands  of  the  United  States,  prescribed  by  the  Commissioner  of  the  General 
Land  Office  ; Clevenger’s  Government  Surveying  ; Burt’s  Solar  Compass. 


BOOK  VI. 

TRIGONOMETRICAL  SURVEYING. 


SECTION  I. 

MAKING  THE  SURVEY. 

241.  Trigonometrical  Surveying,  or  Triangulation,  is 

the  method  of  determining  the  position  of  points  on  the  surface 
of  the  earth  by  the  application  of  the  principles  of  Trigonometry 
— each  successive  point  being  determined  by  the  intersection  of 
two  lines  which  make  known  angles  with  a given  line.  It  may 
be  used  in  small  or  extensive  surveys ; it  does  not  necessarily 
take  into  account  the  curvature  of  the  earth,  though  always  used 
in  the  great  surveys,  such  as  the  U.  S.  Coast  Survey,  in  which 
that  is  considered. 

242. *  In  the  construction  of  a true  map  of  any  large 
territory,  three  things  are  necessary.  The  first  of  these  is  to 
ascertain  the  exact  relation  to  each  other,  as  to  distance  and 
direction,  of  the  leading  features  of  the  country  ; selecting  such 
high  points  as  may  be  seen  from  the  greatest  distance,  and  con- 
necting the  whole  by  a chain  of  triangles  of  the  largest  dimen- 
sions practicable,  and  of  the  forms  most  convenient  for  accuracy 
of  computation. 

The  series  must  rest  upon  a base  of  which  the  geographical 
position  has  been  established  with  mathematical  accuracy. 
The  points  which  form  the  several  angles  of  the  chain  will  thus 


Report  of  N.  Y.  State  Survey,  1878. 


SEC.  I.] 


MAKING  THE  SURVEY. 


213 


be  fixed  with  equal  exactness  as  points  of  the  general  surface  of 
the  earth.  With  the  sides  of  this  principal  chain  of  triangles 
as  bases,  a net  of  smaller  triangles  is  then  to  be  constructed 
occupying  the  interior  of  the  larger,  and  resting  their  angles 
upon  the  most  conspicuous  objects  observable  from  the  principal 
stations.  And  within  these  still  a third  series  is  to  be  formed, 
connecting  as  many  of  the  less  important  points  within  those  of 
the  second  order  as  the  objects  of  the  survey  may  seem  to  require. 
Every  angle  of  every  series  of  these  triangles  marks  a geograph- 
ical point  determined  with  the  same  degree  of  precision  as  that 
of  the  original  base  from  which  the  triangulation  began.  The 
base  on  which  the  New  York  State  Survey  rests  is  a side 
23  miles  long  of  one  of  the  large  triangles  of  the  United  States 
Coast  Survey,  a magnificent  work  which  twenty  years  since 
pushed  its  operations  to  the  head  of  tide-water  upon  the  Hudson 
River.  The  bases  of  the  United  States  Coast  Survey  itself  are 
lines  of  some  miles  in  length,  determined  in  position  by  astro- 
nomical observation,  and  actually  measured  on  the  ground  from 
end  to  end,  by  means  of  apparatus  of  extreme  delicacy,  con- 
structed especially  for  the  purpose. 

243.  The  specimen  of  tri angulation,  Fig.  118,  is  from  a 
survey,  made  by  Professor  Rees  of  Columbia  College,  of  Otsego 
Lake,  N.  Y.,  for  the  N.  Y.  State  Survey,  and  will  serve  to  illus- 
trate the  method  of  making  and  plotting  such  a survey. 

244.  Before  commencing  a trigonometrical  survey,  an  ex- 
amination of  the  entire  territory  should  be  made  for  the  purpose 
of  selecting  a base  line  and  proper  points  for  stations ; this 
examination  should  be  more  or  less  elaborate  according  to  the 
nature  and  extent  of  the  survey. 

The  proper  distribution  and  combination  of  the  triangles,  so 
as  to  adapt  them  to  the  survey  in  band,  require  great  judgment 
and  care,  and  the  selection  of  proper  trigonometrical  points  is  a 


Fig.  118. 


SEC.  I.] 


MAKING  THE  SURVEY. 


215 


very  important  part  of  the  preliminary  operations.  The  selec- 
tion should  be  so  made  that  each  point  may  command  a view  of 
the  greatest  practicable  number  of  surrounding  trigonometrical 
point  objects,  and  that  each  angle  at  the  point  shall  be  as  near  as 
may  be  G0°.  A triangle  which  has  an  obtuse  or  a very  acute 
angle  will  experience  a greater  change  of  form,  for  a given  error, 
than  one  which  is  nearly  equilateral ; and  since  the  accuracy  of 
each  triangle  depends  upon  the  preceding  ones,  it  is  evident  that 
the  introduction  of  a single  “ill-conditioned”  triangle  might 
vitiate  the  whole  survey.  No  angle  less  than  30°  or  more  than 
120°  should  be  used;  and  even  such  angles  should  not 
be  admitted  when  the  locality  can  be  so  chosen  as  to  pre- 
vent it, 

245.  If  the  tri angulation  is  to  be  over  a limited  extent  of 
country  which  has  already  been  covered  by  a net-work  of 
Primary  Triangles,  a side  of  one  of  these  triangles  should  be 
used  as  a base.  It  is  never  good  practice  to  measure  a base  line, 
when  a side  of  a triangle  of  a previous  survey  is  available  ; but  if 
no  such  side  can  be  obtained,  then  the  selection  of  a proper  site 
for  a base-line  forms  one  of  the  first  objects  of  the  preliminary 
reconnaissance.  It  should,  if  possible,  be  fixed  on  an  open  plain, 
free  from  surface  encumbrance  or  freed  from  such.  It  must  be 
so  chosen  that  the  surrounding  signals  may  be  distinctly  seen 
from  its  extreme  points ; and  hence  those  signals  which  mark 
points  of  the  adjacent  triangulation,  should  be  selected  with 
reference  to  the  base.  The  length  of  the  base  should  be  suited 
to  the  magnitude  of  the  survey. 

246.  In  measuring  a base-line,  every  possible  precaution 
should  be  taken  to  insure  accuracy.  The  line  measured  should 
be  straight,  to  effect  which  it  should  be  ranged  out  with  the 
transit.  The  ends  of  the  base  should  be  marked  by  a stone  sunk 
in  the  ground,  with  a copper  bolt  let  into  it  and  the  exact  point 


216 


ELEMENTS  OF  SURVEYING. 


[ROOK  VI. 


of  beginning  and  ending  fixed  by  the  intersection  of  two  lines 
cut  into  the  bead  of  the  bolt. 

The  measurement  may  be  made  with  steel  tape  or  rods.  If  a 
tape  is  used,  it  should  be  carefully  drawn  out  each  time  to  its 
standard  length,  and  should  be  compared  with  a standard  both 
before  and  after  measurement,  and  correction  made  for  its 
variation,  if  any,  from  standard.  The  mean  of  several  measure- 
ments should  be  taken  for  the  correct  measurement.  If  the 
measurement  has  been  made  on  an  incline,  instead  of  on  a level, 
the  measured  distance  should  be  reduced  to  the  horizontal 
distance  by  multiplying  the  inclined  distance  measured  by  the 
cosine  of  the  angle  of  inclination. 

247.  For  a description  of  the  base-line  apparatus  used  in  the 
U.  S.  Coast  and  Geodetic  Survey,  see  Reports  of  that  Survey  for 
1854,  1857,  and  1880. 

248.  The  alignment  of  the  measuring  tape,  or  rods,  both 
vertically  and  horizontally,  or  in  the  line  of  the  slope  if  the 
measurement  be  not  horizontal,  is  of  the  greatest  importance, 
since  there  is  no  compensation  of  errors,  a faulty  alignment 
always  resulting  in  a measured  length  greater  than  the  true 
length. 

249.  Having  carefully  measured  the 
base,  it  is  then  necessary  to  reduce  the 
measurement  to  the  sea  level. 

Let  L — measured  base. 

Let  l = reduced  base. 

Let  R = radius  of  earth. 

Let  h = average  height  of  measured 
base  above  sea  level ; 

then, 


l : L ::  R : R+h 


SEC.  I.] 


MAKING  THE  SURVEY. 


217 


But  li  is  so  small  as  compared  with  R that  we  may,  without 
sensible  error,  make  R + h — R j whence  we  have, 

250.  The  trigonometrical  stations  are  marked  by  signals, 
which  may  be  constructed  in  a great  variety  of  ways,  depending 
upon  the  locality  of  the  stations  and  the  lengths  of  the  sides  of 
the  triangles. 

Sometimes  a signal  has  to  be  raised  above  the  level  of  the 
adjacent  country,  in  which  case  it  is  constructed  of  timbers,  and 
upon  the  apex  is  placed  a vertical  staff  bearing  a flag.  The 
exact  trigonometrical  point  is  determined  by  a plumb-line 
suspended  from  the  apex  of  the  signal. 

A temporary  signal  may  be  constructed  with 
three  or  four  pieces  of  scantling  framed  and 
braced,  as  shown  in  the  annexed  figure,  with  a 
short  pole  projecting  from  the  apex.  The  plumb 
determines  the  point  B , which  is  the  exact 
trigonometrical  point  over  which  the  theodolite  is 
to  be  placed.  Where  the  sides  of  the  triangles 
are  not  very  great,  a pole,  planted  vertically  and 
surmounted  by  a flag,  will  answer  as  a signal. 

In  order  to  distinguish  the  different  signals,  the  flags  which 
they  bear  should  be  different  from  each  other.  They  may  be 
formed  by  arranging  stripes  of  white  and  red,  according  to  some 
prearranged  plan,  and  the  flags  of  the  different  stations  should 
be  entered  in  a book.  For  the  purpose  of  future  reference, 
the  trigonometrical  point  at  each  station,  as  B , should  be  indi- 


Fig.  120. 


218 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


cated  by  a permanent  mark.  If  the  point  falls  upon  a rock,  a 
hole  may  be  drilled  to  show  the  locality  ; or  if  not,  a mark-stone 
may  be  sunk  under  the  point,  deep  enough  to  be  beyond  the 
reach  of  accident.  A record  of  the  monument  should  be  pre- 
served, together  with  its  reference  to  some  of  the  permanent 
objects  in  the  neighborhood. 

251.  A Heliotrope  is  necessary  in  long  sights,  and  is  always 
of  great  service  in  short  sights  in  directing  the  observer  to  the 
station-mark.  It  consists  essentially  of  a small  mirror,  so 
directed  by  an  assistant  as  to  throw  a beam  of  sunlight  into  the 
telescope  of  the  distant  observer.  Let  a silvered  glass,  about  3 
inches  square,  be  mounted  on  a board  in  a manner  similar  to  the 
telescope  mount  of  a transit  so  as  to  have  a motion  about  a 
horizontal  axis  and  at  the  same  time  about  a vertical  axis ; in 
front  of  this,  at  a distance  of  two  or  three  feet,  mount  a board 
with  a hole  in  it,  across  which  hole  threads  are  to  be  stretched 
at  right  angles  to  each  other,  and  adjust  this  hole  over  the  station 
by  a plumb-line.  At  the  centre  of  the  back  of  the  mirror  scrape 
away  the  silver,  making  a small  sight-hole;  if  now  an  assistant, 
sighting  through  the  hole  in  the  mirror,  moves  it  so  that  the 
cross  threads  come  in  line  with  the  distant  station,  it  will  be  easy 
to  keep  the  beam  upon  the  observer  by  properly  inclining  and 
revolving  the  mirror.  As  the  light  reflected  from  so  large  a 
mirror  would  be  too  intense  to  observe  with  the  telescope,  it  is 
necessary  to  make  the  cross-thread  hole  in  the  board  quite  small, 
not  more  than  \ inch  diameter  for  distances  not  exceeding  5 
miles,  about  one  inch  diameter  for  distances  of  ten  miles,  and 
so  on. 

A small  pocket  mirror  will  be  found  very  useful  as  a means 
of  telegraphing  instructions,  by  combinations  of  flashes  according 
to  a system  previously  agreed  upon.  It  can  be  directed  to  the 
observing  station  (or  to  the  observed  station)  with  sufficient 


SEC.  I.] 


MAKING  THE  SURVEY. 


219 


accuracy  for  signalling,  by  setting  a vertical  staff  in  line  with  the 
distant  station  and  causing  flashes  to  travel  up  and  down  the 
staff. 

252.  The  extent  of  the  survey,  and  the  standard  of  accuracy 
to  which  the  results  are  required  to  conform,  must  determine  the 
size  and  perfection  of  the  instrument  to  be  employed  in  the 
measurement  of  angles.  The  angles  of  the  primary  triangles  of 
the  United  States  Coast  and  Geodetic  Survey  are  measured  with 
theodolites,  whose  horizontal  circles  are  24  or  30  inches  in 
diameter;  and  to  eliminate,  as  much  as  possible,  every  source  of 
error,  great  numbers  of  observations  are  made  at  each  station, 
the  readings  being  made  on  different  points  of  the  arc  by  different 
verniers.  Usually  from  40  to  60  observations  are  made  for  each 
angle — one  measurement,  with  the  telescope  direct,  and  one  with 
it  reverted,  constituting  a complete  observation.  With  these 
precautions,  it  has  been  found  that  the  error  in  a primary 
triangle  (where  the  sum  of  its  three  angles  has  been  compared 
with  180°),  has  fallen  much  within  3 seconds.  The  error  of 
3 seconds  has  been  adopted  as  the  highest  admissible  limit  of 
error  in  such  triangles. 

253.  The  theodolite  does  not  differ  essentially  in  the 
principles  of  its  construction  and  use  from  the  transit,  which  has 
already  been  described.  It  is  fitted  with  many  appliances,  for 
accuracy  in  the  observation  and  the  reading  of  angles,  which  it 
is  unnecessary  to  describe  here.  Fig.  121  is  a representation  of 
the  8 to  12  inch  theodolite  used  in  the  U.  S.  Coast  and  Geodetic 
Survey,  taken  from  the  Report  of  that  Survey  for  1880. 


22  0 


ELEMENTS  OF  SURVEYING, 


[BOOK  VI, 


SEC.  I.] 


MAKING  THE  SURVEY. 


221 


Fig.  122. 


254.  To  illustrate  the  principle  of  repetition  in  the  measure- 
ment of  angles,  suppose  the  0 of  the  vernier  to  coincide  with  the 
0 of  the  limb,  and  the  telescope  to  be  directed,  from  the  station 
A , Fig.  122,  upon  one  of  the  objects,  as  the  signal  at  B. 
Clamp  the  limb  and,  unclamping  the  vernier  plate,  direct  the 
telescope  on  the  second  object,  as  the  signal  at  E.  If  we  now 
clamp  the  vernier  plate  and,  unclamping  the  limb,  direct  the 
telescope  on  the  signal  at  B,  the  line  (0°,  180°)  of  the  limb  will 
make  with  AB  an  angle  equal  to  BAE.  Again  clamp  the 
limb  and,  unclamping  the  vernier  plate,  direct  the  telescope 
on  the  signal  at  E.  The  reading  will  evidently  be  equal  to 
twice  the  angle  BAE\  and  if  we  repeat  the  operation,  the 
reading  will  be  three  times  the  angle,  and  so  on.  After  ten 
repetitions,  if  we  add  360°  each  time  the  0 of  the  vernier 
passes  the  0 of  the  limb,  the  final  reading  will  be  ten  times  the 


222 


ELEMENTS  OF  SUItY EYING. 


Thook  VI. 


angle  BAE,  affected  with  the  joint  errors  of  the  ten  observations, 
and  one-tenth  of  this  will  be  the  reading  required  to  a greater 
degree  of  accuracy  than  could  probably  be  attained  by  a single 
observation. 

255.  The  method  of  reading  angles  and  recording  notes  is 
as  follows: 

(1.)  First  mark  one  vernier  A and  the  other  B by  pasting 
these  letters  upon  the  vernier  plate.  Set  vernier  A at  0°  and 
clamp  it.  Direct  the  cross-wires  to  the  left  of  station  B and 
then  by  a careful  movement  to  the  right  bring  the  sight 
nearly  upon  station  B,  being  careful  not  to  pass  the  station , 
and  perfect  the  bisection  by  the  lower  tangent  screw.  Read 
both  verniers,  reading  B to  minutes  and  seconds  only,  and 
make  the  entry  as  in  the  table  subjoined.  Now  loosen 
the  vernier  clamp  and  turn  carefully  to  the  right  till  nearly 
upon  station  E,  completing  the  bisection  by  the  vernier 
tangent-screw.  Now  make  a “ check  ” mark  in  column 
“vernier  A ,”  as  in  table.  Next  loosen  the  lower  clamp  and  turn 
the  telescope  to  the  right  till  station  B is  nearly  reached,  and 
repeat  the  previous  operations,  making  “ check”  in  “vernier  A ” 
column  as  before.  In  like  manner  repeat  the  operations  for  the 
third  time,  and  enter  the  final  reading  upon  station  E,  as  in  the 
table. 

If  the  motion  of  vernier  A has  been  fonvard,  in  the  direction 
of  the  graduation  of  the  limb,  the  first  reading  must  be  marked 
negative,  as  in  the  table ; when  the  motion  of  vernier  A is  back- 
ward, the  final  reading  must  be  marked  negative.  In  the  case 
just  described  the  motion  has  been  forward,  therefore  from  the 
mean  of  the  last  readings  (217°  15'  15"),  we  subtract  the  mean 
of  the  first  readings  (O'  10"),  and  divide  the  result  by  the 
number  of  repetitions  (3)  obtained  for  the  mean  angle  72° 
25'  i".7. 


SEC.  I.] 


MAKING  THE  SURVEY. 


223 


(2.)  Set  vernier  A at  180°,  and  having  turned  the  telescope 
to  the  right  of  E , make  three  repetitions  of  the  angle  subtended 
by  EB,  working  from  right  to  left.  In  this  case  vernier  A will 
pass  over  the  360°  mark,  which  fact  is  noted  in  the  table  under 
the  first  “ vernier  B ” reading.  As  vernier  A worked  backward, 
the  last  reading,  that  taken  upon  B,  must  be  negative. 

(3.)  Revolve  the  telescope  upon  its  horizontal  axis,  and  turn 
the  telescope  about  the  vertical  axis  towards  B.  Set  vernier  A 
at  90°  and  repeat  three  times  from  B to  E. 

(4.)  Set  vernier  A at  270°  and  repeat  three  times  from 
E to  B. 

The  final  mean  of  the  four  angles  is  72°  24'  59".  2. 

Both  verniers  are  read  to  eliminate  error  of  centering  of 
graduated  limb. 

Repetitions  are  made  to  eliminate  errors  of  graduation. 

Readings  from  left  to  right , and  from  right  to  left , are  taken 
to  counteract  errors  due  to  torsion,  and  personal  error  of 
bisection. 

The  telescope  is  reversed  on  its  horizontal  axis  to  eliminate 
error  due  to  inequality  of  axis  points,  and  error  of  colli mation. 


224 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


Angles  taken  at  Station  A,  July  17,  1883. 


Sta. 

Obs. 

Vernier  A. 

B. 

± 

Calculation. 

Mean  Ang. 

B 

0°  0'  0" 

0'  20” 

— 

217°  15'  15" 

E 

V V V 

217°  15'  10" 

15'  20" 

0 10 
3)217  15  5 

72°  25'  1”.7 

E 

180°  0'  0" 

O'  10” 

180°  0'  5" 

3G0° 

360 

540  0 5 

322  45  15 

B 

322°  45'  10” 

45'  20” 

— 

3)217  14  50 

72°  24'  56”.7 

B 

90°  0'  0” 

0'  0” 

— 

307°  14'  50” 

E 

vW  V 

307°  14'  40” 

15'  0” 

90  0 0 

3)217  14  50 

72°  24'  56”.7 

E 

270°  0'  0" 

0'  20” 

270°  0'  10” 

B 

52°  45'  0" 

45'  10” 

_ 

52  45  5 

3)217  15  5 

72°  25'  1”.7 

4)  99'  5G".8 

Final  mean. 

72°  24'  59".2 

SEC.  I.] 


MAKING  THE  SURVEY. 


225 


256.  If  several  stations  can  be  read  in  succession,  a simple 
modification  of  the  method  is  practicable.  Suppose  the  observing 
station  to  be  E of  Fig.  122,  and  we  wish  to  read  A,  H,  B , C,  &c. 
Sight  A with  the  vernier  set  at  0°  ; unclamp  the  vernier  and 
turn  to  the  right  till  H is  bisected  and  enter  the  reading  opposite 
H in  the  table  below  ; unclamp  the  vernier  and  sight  B , and 
enter  its  reading;  continue  the  motion  to  the  right,  reading  each 
station  in  turn.  Suppose  C to  be  the  station  on  the  extreme 
right ; now  set  the  vernier  at  180°  and  work  from  Cy  around  to 
the  left,  to  A,  entering  the  readings  in  the  table  in  a reverse 
order,  from  the  bottom  to  the  top  of  the  page.  Reverse  the 
telescope,  and  with  90°  as  the  first  reading  work  from  A to  C \ 
and  then  with  270°  as  the  first  reading  work  from  C to  A.  The 
notes  are  given  below ; the  readings  on  station  A subtracted  from 
the  corresponding  readings  on  H,  B , and  C , give  the  angular 
distances  of  these  points  from  the  line  EA. 


Angles  taken  at  Station  E. 


Sta. 

Vernier  A. 

B. 

Mean  Read. 

Angles  with  Line 
EA. 

Mean  Angles  with 
EA. 

A 

0° 

0' 

0" 

0' 

20" 

0° 

0' 

10" 

72 

0 

10 

0 

0 

72 

0 

5 

90 

0 

0 

0 

10 

90 

0 

5 

162 

0 

20 

0 

10 

162 

0 

15 

H 

40° 

O' 

10" 

0' 

20" 

o 

O 

0' 

15" 

o 

O 

0' 

5" 

112 

0 

0 

0 

10 

112 

0 

5 

40 

0 

0 

130 

0 

10 

0 

30 

130 

0 

20 

40 

0 

15 

202 

0 

10 

0 

10 

202 

0 

10 

39 

59 

55 

o 

O 

0'  3".75 

B 

66° 

30' 

0" 

30' 

20" 

66° 

30' 

10" 

66° 

30' 

0" 

138 

29 

40 

30 

20 

138 

30 

0 

66 

29 

55 

156 

29 

40 

30 

10 

156 

29 

55 

66 

29 

50 

228 

30 

0 

30 

0 

228 

30 

0 

66 

29 

45 

66° 

29'52".5 

ELEMENTS  OF  SURVEYING. 


[ROOK  VI. 


22  6 


Sta. 

Vernier  A. 

B. 

Mean  Read. 

Angles  with  Line 
EA. 

Mean  Angles  with 
EA. 

c 

o 

00 

o 

O' 

0" 

0'  10" 

108° 

0' 

5" 

107°  59'  55" 

180 

0 

0 

0 20 

180 

0 

10 

108  0 5 

198 

0 

10 

0 10 

198 

0 

10 

108  0 5 

270 

0 

0 

0 10 

270 

0 

5 

107  59  50 

107°  57'  58".75 

Having  found  by  these  methods  the  angles  of  any  triangle, 
one  of  its  sides  being  the  base-line  or  a known  side  of  a triangle 
already  computed,  we  can  find  the  sides  of  the  new  triangle. 

First  subtract  the  sum  of  the  angles  from  180°  and  apply  of 
the  error  to  each  angle;  next  treat  the  triangle  as  a plane 
triangle  and  compute  the  two  required  sides. 

257.  The  spherical  excess  in  triangles  of  a Primary  System 
is  seldom  more  than  about  6",  and  triangles  whose  sides  are  not 
more  than  ten  miles  long  may  be  regarded  as  plane  triangles 
without  sensible  error. 

258.  It  sometimes  happens  that  a steeple,  tower,  or  other 
prominent  object  must  be  used  as  a station,  and  in  most  cases  it 
is  impossible  to  set  the  instrument  over  the 
centre  of  such  station.  In  such  cases  a 
“ reduction  to  the  centre”  is  necessary. 

Let  0 be  the  position  of  the  instrument, 

C the  centre  of  a circular  tower  which  marks 
the  station,  and  DGG  the  desired  angle,  D 
being  the  right-hand  object  and  G the  left- 
hand  one.  Measure  (in  some  one  of  the 
many  ways  for  indirect  measurement)  the 
distance  OG  = r,  and  take  the  angle  GOG  — y9 
always  measuring  the  angle  y (called  the 
angle  of  direction)  from  the  left-hand  object, 


Iig.  123. 


SEC.  I.] 


MAKING  THE  SURVEY. 


22? 


G,  and  estimating  it  towards  the  left,  from  0°  to  360°,  as  in  the 
figure.  The  angle  I being  exterior  to  the  triangle  DOI  we  have 
I=0-\-a , and  from  the  triangle  GIG  we  have  / = C -f-  (3, 
from  which  we  deduce 

C=0  + (a—P)  (11 

We  now  need  to  determine  the  correction  (a  — /3’).  In 

the  triangle  DOC  we  have  (calling  the  side  DC,  D)  sin  a — 

rsin(0  + y)/A  , ..  , 

-jj — (Art.  32),  or  as  a is,  in  practice,  always  very  small 

we  may  substitute  for  its  sine  its  value  in  seconds,  making  sin 

a — (a)"  sin  T',  which  substituted  above  gives  (a)"—  — 

° D sin  1 

In  like  manner  (calling  the  side  GC,  G)  we  have  (13)"  = 
v sin  ?/ 

frsiu~l'7,  Substituting  these  values  in  formula  (1),  we  obtain 


C = 0 + 


r sin  (O+y)  r sin 

D sin  1"  G sin 


in  y_) 
in  1"  j 


(2) 


This  formula  is  general  if  care  is  taken  to  estimate  the  angle 
y with  reference  to  the  positions  of  the  stations  D,  G,  and  C,  as 
above  directed,  the  signs  of  the  trigonometric  functions  also 
being  observed  (see  Davies’  Leg.,  Trig.  Art.  58). 

The  angle  0 is  measured  in  the  usual  manner.  The  angle  y, 
in  the  case  supposed,  is  obtained  by  measuring  the  angle  which 
each  tangent  to  the  tower  through  0 makes  with  the  line  OG, 
and  taking  their  half  sum. 

In  formula  (2)  the  sides  D and  G are  unknown.  In  triangles 
whose  sides  are  not  less  than  3000  feet,  r being  relatively  very 
small,  it  will  be  sufficient  to  calculate  these  sides  from  the  known 
side  and  the  uncorrected  observed  angles,  and  substitute  such 
computed  values  in  formula  (2). 

When  the  sides  are  less  than  3000  feet,  r being  relatively 
large,  the  values  of  D and  G are  obtained  by  two  approximations. 
First  plot  a triangle  O'  D'  G'  upon  as  large  scale  as  practicable, 


228 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


using  the  known  side  and  the  observed  angles;  then  plot  the 
angle  y , and  lay  off  O' C = nr,  n being  either  10,  20,  30,  &c.,  so 
that  nr  may  measure  between  ^ and  ^ of  the  length  of  the 
shorter  of  the  two  plotted  sides  O' D'  and  O' O',  in  which  case 

O'  D'  C 

the  plotted  angle  O’ DC'  will  be  equal  to  nxa,  or  a = — : — , 

w 

Qt  Q!  (J! 

and  (3  = — • These  values  substituted  in  formula  (1) 

n v 7 

give  a value  of  C which  may  be  used  in  the  computation  of  the 

sides  D and  G,  and  these  computed  values  of  D and  G must  then 

be  substituted  in  formula  (2). 

259.  An  error  of  observation  may  also  arise  from  the 
unequal  illumination  of  the  face  of  the  object  presented  to  the 
observer,  called  error  due  to  phase ; thus,  if  an 
observer  at  A sight  a tower  at  D , the  sun  being 
in  the  direction  8,  he  will  direct  his  telescope  to 
d,  the  middle  of  the  illuminated  portion,  instead 
of  to  D.  This  error,  and  also  that  due  to 
irradiation  in  the  case  of  ordinary  signals,  may 
be  avoided  by  observing  when  the  sun  is 
obscured,  or  by  throwing  a shadow  upon  rods  and 
other  commonly  used  signals.  It  is  hardly  worth  while  to  intro- 
duce a formula  for  correction  of  phase  in  this  connection. 

(For  the  methods  used  in  the  work  of  the  U.  S.  Coast  and 
Geodetic  Survey,  see  “ Field  Work  of  the  Triangulation,”  1877, 
by  Richard  D.  Cutts,  assistant.) 

260.  When,  in  connection  with  a trigonometrical  survey  on 
shore,  a harbor  is  to  be  surveyed  (see  Fig.  122),  for  the  purpose 
of  ascertaining  the  channels,  their  depth  and  width,  the  positions 
of  shoals,  and  the  depth  of  water  thereon,  other  means  must  be 
used,  and  other  examinations  made,  in  addition  to  those  already 
described. 


Fig.  124. 


SEC.  I.] 


MAKING  THE  SURVEY. 


229 


Let  buoys  be  anchored  on  the  principal  shoals  and  along  the 
edges  of  the  channel ; and  using  any  one  of  the  lines  already 
determined  as  a base,  let  the  angles  subtended  by  lines  drawn 
from  its  extremities,  to  the  buoys  respectively,  be  measured  with 
the  theodolite.  Then,  there  will  be  known,  in  each  triangle,  the 
base  and  the  angles  at  the  base,  from  which  the  distances  to  the 
buoys  are  easily  found  ; and  hence,  their  positions  become 
known. 

Having  made  the  soundings  and  ascertained  the  exact  depth 
of  the  water  at  each  of  the  buoys,  several  points  of  the  harbor 
are  established,  at  which  the  precise  depth  of  the  water  is 
known ; and  by  increasing  the  number  of  the  buoys,  the  depth 
of  the  water  can  be  found  at  as  many  points  as  may  be  deemed 
necessary. 

261.  If  a person  with  a theodolite,  or  transit,  be  stationed  at 
each  extremity  of  the  base-line,  it  will  not  be  necessary  to 
establish  buoys.  A boat,  provided  with  an  anchor,  a sounding- 
line, and  a signal-flag,  has  only  to  throw  the  anchor,  hoist  the 
signal-flag,  and  make  the  sounding,  while  the  persons  at  the 
extremities  of  the  base-line  measure  the  angles.  From  these 
data,  the  precise  place  of  the  boat  can  be  determined. 

262.  There  is  another  method  of  determining  the  places  at 
which  the  soundings  are  made,  that  admits  of  great  despatch, 
and  which,  if  the  observations  are  made  with  care,  alfords  results 
sufficiently  accurate. 

Having  established,  trigonometrically,  three  points  which  can 
be  seen  from  all  parts  of  the  harbor,  and  having  provided  a 
sextant,  let  the  sounding  be  made  at  any  place  in  the  harbor,  and 
at  the  same  time  the  three  angles  subtended  by  lines  drawn  to 
the  three  fixed  points,  measured  with  the  sextant. 

The  problem,  to  find,  from  these  data,  the  place  of  .the  boat 
at  the  time  of  the  sounding,  is  the  same  as  Example^,  Art.  203. 


230 


ELEMENTS  OF  SURVEYING. 


[ROOK  VI. 


It  is  only  necessary  to  measure  two  of  the  angles,  but  it  is 
safest  to  measure  the  third  also,  as  it  affords  a verification  of 
the  work. 

The  great  rapidity  with  which  angles  can  be  measured  with 
the  sextant  by  one  skilled  in  its  use,  renders  this  a most  expe- 
ditious method  of  sounding  and  surveying  a harbor. 

For  description  of  the  sextant  and  its  method  of  use,  see 
Appendix  B. 

263.  There  is  yet  another  method  of  finding  the  soundings, 
which,  although  not  as  accurate  as  those  already  explained, 
will,  nevertheless,  afford  results  approximating  nearly  to  the 
truth.  It  is  this: — Let  a boat  be  rowed,  with  uniform  speed, 
across  the  harbor,  from  one  extremity  to  the  other  of  any  of  the 
lines  determined  trigonometrically.  Let  soundings  be  made 
continually,  and  let  the  precise  time  of  making  each  be  carefully 
noted.  Then,  knowing  the  length  of  the  entire  line,  the  time 
spent  in  passing  over  it,  as  also  the  time  of  making  each  of  the 
soundings,  we  can  easily  find  the  points  of  the  line  at  which  the 
several  soundings  were  made ; and  hence,  the  depth  of  water  at 
those  points  becomes  known. 

264.  If  a person  stationed  on  shore  with  a theodolite  or 
transit,  takes  the  bearing  of  the  boat,  at  every  second  or  third 
sounding,  determined  by  hoisting  a flag,  it  will  fix  the  positions 
of  the  soundings  with  great  accuracy.  Soundings  may  thus  be 
made  along  any  number  of  known  lines,  and  a comparison  of 
the  depths  found,  on  different  lines,  at  or  near  their  points  of 
intersection,  will  show  with  what  degree  of  accuracy  the  work 
has  been  done. 

Sounding-lines  should  be  made  of  stroug  cord,  and  divided 
into  feet  or  fathoms,  by  different  colored  rags  or  other  marks. 
The  lead  is  shaped  like  the  frustum  of  a cone,  with  the  base  B 
hollowed  or^,  to  hold  some  grease.  The  land  or  mud  of  the 


SEC.  I.J 


MAKING  THE  SURVEY. 


231 


bottom  adheres  to  the  grease,  and  thus  shows  the  nature  of  the 
bottom,  which  should  be  entered  in  the  field- 
book,  and  laid  down  upon  the  map.  As  the 
cord  is  liable  to  change  its  length,  it  should 
be  compared,  from  time  to  time,  with  some 
standard.  In  tide-waters,  the  exact  time  of 
each  sounding  is  to  be  noticed,  and  an 
assistant  should  note  the  height  of  the  tide  at 
regular  intervals,  upon  a tide-gauge.  The 
tide  gauge  is  permanently  placed  at  some 
convenient  point  of  the  harbor,  and  its  0 
point  is  referred,  by  means  of  a spirit-level,  to  some  fixed  bench- 
mark, on  a level  with  mean  low-water  mark,  to  which  all  the 
soundings  must  be  reduced. 

265.  Having  plotted  the  work  done  with  the  theodolite  or 
transit,  as  also  the  outline  of  the  harbor  traced  with  the  compass, 
it  remains  to  delineate  the  bottom  of  the  harbor;  and  this  is  done 
by  means  of  horizontal  curves,  hereafter  explained  (Art.  332), 
which  are  used  to  represent  broken  or  undulating  ground. 

Let  the  plane  of  reference  be  taken  through  low-water  mark, 
or  to  coincide  with  the  surface  of  the  water  at  low  tide.  The 
accuracy  with  which  the  bottom  of  the  harbor  is  to  be  delineated, 
will  guide  us  in  fixing  the  distance  between  the  horizontal  planes 
of  section. 

The  first  horizontal  plane  should  be  passed  at  a distance 
below  the  shallowest  point  that  has  been  sounded,  equal  to  the 
number  of  feet  fixed  upon  for  the  distance  between  the  planes 
of  section  ; and  the  curve,  in  which  it  intersects  the  bottom  of 
the  harbor,  determined  as  in  Art.  335.  And  similarly,  for  the 
other  horizontal  planes  of  section. 

Having  thus  delineated  the  bottom  of  the  harbor,  and  noted 
on  the  map  the  distance  of  each  intersecting  plane  below  the 


232 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


plane  of  reference,  let  such  lines  be  drawn  as  will  indicate  the 
channels,  shoals,  sunken  rocks,  and  direction  of  the  current. 

In  the  example  given  in  Fig.  122,  soundings  have  been 
made  in  three  directions,  from  the  sand-bar  in  the  harbor,  and 
also  from  the  rocky  shore  across  to  the  light-house. 

266.  When  a large  extent  of  territory,  or  a long  line  of  sea- 
coast  is  to  be  surveyed,  it  becomes  necessary  to  consider  the 
curvature  of  the  earth’s  surface ; this  branch  of  surveying  is 
called  Geodetic  surveying. 

The  operations  necessary  to  the  successful  execution  of  a 
Geodetic  Survey,  require  the  minutest  attention,  and  when 
performed,  numerous  corrections  are  to  be  applied  to  the  meas- 
ured lines  and  angles,  on  account  of  the  various  causes  of  error 
incident  to  such  operations. 

To  investigate  those  causes  of  error,  and  to  deduce  rules  for 
correcting  the  errors,  in  all  cases,  would  exceed  the  limits  of 
this  treatise.  We  have,  therefore,  attempted  nothing  more 
than  an  outline  of  the  operations  in  a trigonometrical  survey, 
in  which  the  Plane-Table  and  Compass  are  used  in  con- 
nection with  the  Theodolite  or  Transit,  and  in  which  the 
curvature  of  the  earth  is  not  considered. 


SEC.  II.] 


FILLING  UP  THE  SURVEY. 


233 


SECTION  II. 

FILLING  UP  THE  SU  RVEY. 

After  the  triangulation  is  completed,  the  interior  may  dp 
filled  up  by  the  aid  of  the  Compass,  or  Plane-Table,  or  both. 

By  the  Compass. 

267.  The  use  of  the  compass,  in  determining  points  and  lines,, 
by  means  of  offsets,  has  been  already  explained  (Art.  123).  We 
will  apply  these  principles  in  the  example  of  the  harbor,  Fig.  122. 

Place  the  compass  at  A..  and  take  the  bearing  of  the  line 
AE,  which  is  S 12°  W. 


Fig.  126. 


Enter  this  bearing  at  A.  Then  measure  along  the  line  AE 
any  distance,  as  A a ecpial  to  130  yards,  and  make  an  offset  to 
the  lake,  which  we  measure  and  find  to  be  50  yards.  Enter 
the  130  in  the  middle  column,  and  as  the  lake  lies  on  the  right 


234  ELEMENTS  OF  SURVEYING.  [BOOK  VI. 

(in  going  from  A to  E ),  we  insert  the  50  in  the  right-hand 
column. 

We  then  measure  along  the  line  AE  to  b,  350  yards  from  A. 
Here  we  make  a second  offset  to  the  lake,  and  find  it  to  be  equal 
to  100  vards.  Having  entered  the  distances  in  the  notes,  we 
measure  to  q,  the  point  where  the  line  AE  touches  the  creek, 
and  we  enter  the  distance  from  A,  415  yards. 

At  cl,  we  lay  off  an  offset  on  the  left,  to  the  pond,  70  yards ; 
at  e,  an  offset  to  the  mouth  of  the  creek,  150  yards ; and  at  E, 
where  the  course  terminates,  an  offset  to  the  lake,  of  1G0  yards. 
The  entire  distance  from  A to  E is  800  yards. 

At  E,  we  take  the  bearing  to  H,  which  is  N 50°  E.  Having 
measured  along  this  line  to  /,  315  yards,  we  make  an  offset  to 
the  pond,  on  the  left,  of  50  yards,  and  to  the  shore,  on  the  right, 
of  90  yards.  Having  entered  these  distances,  we  recommence 
the  notes  at  315,  below,  which  we  suppose  to  be  at  the  bottom 
of  the  second  page.  Having  reached  H,  the  extremity  of  the 
course,  we  enter  the  entire  distance  from  E,  680  yards.  We 
next  take  the  bearing  to  I,  S 52°  E.  We  then  measure  the  dis- 
tances to  m,  n,  p,  and  I,  and  enter  them,  together  with  the 
offsets,  as  in  the  notes. 

It  is  also  well  to  make,  in  the  columns  on  the  right  and  left, 
such  sketches  of  the  ground,  fields,  houses,  creeks,  and  rivers, 
as  will  afford  the  means  of  making  an  accurate  delineation  on 
paper. 

By  the  Plane-Table. 

268.  The  plane-table,  Eig.  127,  consists  of  two  parts:  a 
rectangular  board,  and  a tripod  to  which  it  is  firmly  secured. 

l)i  rectly  under  the  rectangular  board  arc  four  milled  screws 
which  pass  through  sockets  inserted  in  a horizontal  brass-plate; 
these  screws  are  worked  against  a second  horizontal  plate,  for 


SEC.  II.] 


FILLING  UP  THE  SURVEY. 


235 


the  purpose  of  leveling  the  table ; the  table  having  a ball-and- 
socket  motion,  similar  to  the  limb  of  the  transit. 


Fig.  m. 


Between  the  upper  horizontal  plate  and  the  table,  there  is 
a clamp-screw,  similar  to  the  clamp-screw  of  the  transit,  which 
being  loosened,  the  table  can  be  turned  freely  about  its  axis. 
There  is,  also,  a small  tangent  screw,  by  which  the  smaller 
motions  of  the  table  are  regulated,  after  the  clamp-screw  is 
made  fast. 

A set  of  brass  clamps  for  fastening  the  paper  to  the  table ; 
a clamp  with  an  attachment  on  one  side  for  fastening  a plumb- 
line,  and,  on  the  other,  a pin  immediately  over  the  point  of 
attachment  of  the  plumb-line,  for  marking  on  the  paper  a point 


236  ELEMENTS  OF  SURVEYING.  [BOOK  VI. 

directly  over  a station  of  the  ground  ; an  Alidade  and  a Declina- 
tor accompany  the  Plane-Table. 

The  Alidade  is  a brass  ruler,  from  one  to  two  feet  long, 
with  a fiducial  or  true  straight  edge.  On  the  upper  face  of 
the  ruler  are  two  spirit-levels  at  right  angles  to  each  other, 
and,  near  the  middle  of  the  ruler,  a brass  standard  which  carries 
a telescope,  like  the  telescope  of  the  transit,  with  a vertical 
arc  for  measuring  vertical  angles,  and  micrometer  wires  for 
measuring  distances  (see  Art.  206).  The  telescope  is  so  placed 
with  regard  to  the  edge  of  the  ruler  beneath,  that  its  line  of 
collimation,  when  properly  adjusted,  is  parallel  to  that  edge  and 
in  the  same  vertical  plane. 

The  Declinator  is  a metal  box,  containing  a magnetic  needle 
with  a range  of  about  10°  on  each  side  of  the  0.  It  is  used  to 
orient  the  table,  i.  e.,  to  place  it  at  any  point  of  the  field  in  the 
same  position  with  respect  to  the  points  of  the  compass  that  it 
had  at  other  stations  in  the  same  survey ; or,  in  other  words,  to 
place  it  in  such  position  at  a new  station  that  a line  previously 
drawn  upon  the  paper  shall  be  parallel  to  the  line  of  sight  which 
it  represents.  The  orientation  may  be  effected  as  follows: 
While  the  table  is  in  position  for  drawing  the  first  lines  of 
the  survey,  place  upon  it  the  declinator,  with  one  of  the  longer 
edges  of  its  box  set  along  a line  drawn  on  the  paper  for  the 
purpose,  and  note  the  reading  of  the  needle ; when  the  table  is 
removed  to  a new  position,  turn  it  till  the  declinator,  placed 
along  the  line  as  before,  gives  the  same  reading.  Figure  127 
represents  the  plane-table  and  its  accompaniments  as  used  on 
U.  S.  Coast  Survey. 

269.  The  plane-table  is  used  to  determine  the  shorter  lines 
of  a survey  in  extent  and  position. 

Having  placed  a paper  on  the  table,  examine  the  objects  and 
lines  which  are  to  be  determined,  and  select  for  a base  such  a 


SEC.  II.] 


FILLING  UP  THE  SURYEY. 


237 


line  of  the  triangulation  that  most  of  the  objects  can  be  seen 
from  its  extremities.  Then  place  the  plane-table  over  one  ex- 
tremity of  the  base;  make  it  truly  horizontal,  and  turn  it  until 
the  larger  part  of  the  paper  lies  on  the  same  side  of  the  base 
with  the  objects. 

Then,  tighten  the  clamp-screw,  and  mark  with  a pointed 
pin  the  point  of  the  paper  directly  over  the  station,  which  point 
is  determined  most  accurately  by  suspending  a plumb  from 
the  lower  side  of  the  table.  Press  the  pin  firmly  on  this 
point,  bring  the  fiducial  edge  of  the  ruler  against  it,  and  sight 
to  the  other  extremity  of  the  base-line,  and  mark,  with  the  pin 
or  pencil,  the  direction  of  the  line  on  the  paper.  Sight,  in 
like  manner,  to  every  other  object,  and  draw  on  the  paper 
the  corresponding  lines,  numbering  them  from  the  base-line, 
1,  2,  3,  4,  &c. 

Then,  with  a pair  of  dividers,  take  from  the  scale  a certain 
number  of  equal  parts,  to  represent  the  base,  and  lay  off  the 
distance  on  the  base-line  from  the  place  of  the  pin.  Take  up 
the  table,  carry  it  to  the  other  extremity  of  the  base,  and 
place  the  point  of  the  paper  corresponding  to  that  extremity, 
directly  over  it.  Place  the  fiducial  edge  of  the  ruler  on  the 
base-line,  and  turn  the  table,  by  means  of  the  tangent  screw, 
until  the  sights  are  directed  to  the  first  station.  If,  however, 
in  bringing  the  table  to  this  position,  the  corresponding  point 
of  the  paper  has  been  moved  from  over  the  extremity  of  the 
base-line,  move  the  legs  of  the  tripod  until  it  is  brought 
back  to  its  place.  Let  the  table  be  then  leveled,  after  which, 
place  the  ruler  again  on  the  base-line,  and  bring  the  table 
to  its  proper  position,  by  the  tangent-screw,  and  continue  the 
adjustment  until  the  extremity  of  the  base-line,  on  the  paper, 
is  directly  over  the  station,  and  in  the  same  vertical  plane 
with  the  base-line,  on  the  ground.  Then  direct  the  sights  to 


238 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


all  the  objects  sighted  to,  from  the  other  station,  and  mark 
the  lines  1,  2,  3,  4,  &c.,  from  the  base-line,  as  before.  The 
intersections  of  the  corresponding  lines  1,1,  2,2,  3,3,  4,4,  &c., 
determine,  on  the  paper,  the  positions  of  the  several  objects, 
and  a reference  of  these  lines  to  the  scale  of  equal  parts, 
determines  the  true  distances. 

270.  Let  it  be  required,  for  exam- 
ple, to  determine,  by  means  of  the 
plane-table,  the  relative  positions  of 
several  houses. 

From  station  A , and  on  one  of  the 
lines  of  the  triangulation,  as  AB, 
measure  the  base-line  AN,  which  we  will  suppose  equal  to 
300  yards.  Place  the  plane-table  at  A,  and  sight  to  the  corners 
of  the  houses,  and  mark  the  lines  1,  2,  3,  4,  &c.  Then  remove 
the  table  to  N,  and  sight  to  the  same  corners  as  before,  and  draw 
the  lines  as  in  the  figure.  The  points  at  which  they  intersect 
the  corresponding  lines,  before  drawn,  determine  the  corners  of 
the  houses.  The  front  lines  of  the  houses  may  then  be  drawn  on 
the  paper.  Draw  lines  at  right  angles  to  the  front  lines,  and 
on  them  lay  oft  the  depths  of  the  houses,  with  the  same  scale 
as  that  used  for  the  base  line. 

To  find  the  length  of  any  line  drawn  on  the  paper,  as  the 
line  1,  drawn  through  A,  for  example,  place  the  dividers  at  A 
and  extend  them  to  the  other  extremity  of  the  line,  and  then 
apply  the  line  to  the  scale.  The 
length  of  the  line  1 is  equal  to 
108  yards. 

271.  In  th  is  example,  we  de- 
termine from  the  base-line  CD , 
the  positions  of  the  points  F,  E, 
and  II. 


N 1 C 

Fig.  129. 


SEC.  II.] 


FILLING  UP  TIIE  SURVEY. 


239 


Changing  the  Paper. 


272.  When  one  paper  is  filled,  and  there  is  yet  more  work 
to  he  done,  let  the  paper  be  removed,  and  a second  paper  put 
on  the  table ; after  which,  the  table  may  be  used  as  before. 

Now,  in  order  that  the  two  papers  may  be  put  together  and 
form  one  entire  plan,  it  is  necessary  that  two  points,  determined 
on  the  first  paper,  be  also  determined  on  the  second ; and  then, 
by  placing  the  lines  joining  these  points,  one  on  the  other,  all 
the  lines  on  the  two  papers  will  have  the  same  relative  position 
as  the  corresponding  lines  on  the  ground  ; and  the  same  for  as 
many  papers  as  it  may  be  necessary  to  use.  If  different  scales 
are  used,  the  corresponding  points  will  not  join,  and  then  the 
work  must  be  reduced  to  the  same  scale,  before  the  papers  can 
be  put  together. 

In  the  first  example,  the  position  of  the  point  F was  deter- 
mined, in  order  to  unite  the  first  paper  with  the  second. 

In  the  second  example,  we  sighted  from  C and  D,  the  ex- 
tremities of  the  base-line,  to  the  points  N and  F,  determined 
on  the  first  paper ; we  thus  determined  the  line  NF  on  the 
second  paper.  Placing  the  line  NF  of  the  one  paper  on  NF  of 
the  other,  we  have  the  following  plan  : 


^ rr  tt 


A* 


Fig.  130. 


In  this  plan,  all  the  points  and  lines  are  accurately  laid  down. 
Any  number  of  papers  may  be  joined  in  a similar  manner. 


240 


ELEMENTS  OF  SURVEYING. 


[book  VI. 


273.  * 'or  further  description  of  the  plane-table  and  its  use, 
see  “A  Treatise  on  the  Plane-Table  and  its  use  in  Topographical 
Surveying,”  by  E.  Hergesheimer,  U.  S.  Coast  and  Geodetic  Sur- 
vey, Report  for  1880,  Appendix  No.  13. 


SECTION  III. 

PLOTTING  THE  T R I A N G U L A T I 0 N . 

The  sides  of  the  triangles  having  been  completed,  the  work 
may  then  be  plotted,  as  already  explained,  either  by  means 
of  the  circular  protractor,  or  by  the  method  of  chords. 

The  Circular  Protractor. 

274.  This  instrument  consists  of  a brass  circular  limb. 
Fig.  131,  of  about  six  inches  in  diameter,  with  a movable  index 
AB,  having  a vernier  at  one  extremity  A , and  a milled  screw 
at  the  other  extremity  B , with  a concealed  cog-wheel  that  works 
with  the  cogs  of  the  limb,  and  thus  moves  the  index  AB  about 
the  centre  of  the  protractor.  At  the  centre  of  the  protractor 
is  a small  circular  glass  plate,  on  which  two  lines  are  cut;  the 
point  of  their  intersection  is  the  exact  centre  of  the  instru- 
ment. The  limb  is  generally  divided  to  half-degrees;  the  de- 
grees are  numbered  from  0 to  360. 

At  the  0 point,  and  at  the  opposite  extremities  of  the 
diameter  passing  through  that  point,  are  small  lines  on  the 
inner  edge  of  the  limb  ; the  two  extremities  of  the  diameter, 
perpendicular  to  this  latter,  are  designated  in  the  same  way. 

Two  angular  pieces  of  brass,  each  having  a small  and 
sharp  steel  pin  at  its  extremity,  arc  fastened  to  the  index,  and 
revolve  freely  around  the  lines  ab  and  cd.  The  small  screws, 


SEC.  II  I.  J 


PLOTTING  THE  TRIANGULATION. 


a,  b,  c,  and  d,  move  them  in  the  directions  of  the  lines  ab,  cd \ 
fcr  the  purpose  of  bringing  the  steel  pins  exactly  into  the  line 
which  passes  through  the  0 of  the  index  and  the  centre  of 
the  protractor. 

To  adjust  them  to  their  places,  place  the  centre  of  the  pro- 
tractor over  a marked  point,  and  the  0 of  the  index  to  the  0 
of  the  limb.  Then  mark  the  place  of  the  index  by  the  pins; 
after  which,  turn  the  index  180°,  and  see  if  the  pins  will  mark 


242 


elements  of  surveying. 


[hook  vi. 


the  same  points  as  before.  If  they  do,  the  index  is  adjusted ; 
if  they  do  not,  correct  the  error  with  the  screws  a , b,  c,  and  d. 

275.  To  lay  off  an  Angle  with  the  Protractor. — Let  its 

centre  be  placed  over  the  angular  point,  and  the  diameter  pass- 
ing through  0 and  180°,  on  the  given  line.  Turn  the  screw 
that  works  the  index,  until  the  0 of  the  vernier  coincides  with 
the  division  corresponding  to  the  given  angle;  then  let  the 
angular  brass  pieces  be  turned  down ; the  points  dotted  by  the 
steel  pins  will  show  the  direction  of  the  required  line. 

If  this  line  does  not  pass  through  the  angular  point,  the 
pins  are  out  of  place,  and  must  be  re-adjusted. 

276.  First  Method  of  Plotting. — Suppose  it  were  required 
to  make  the  plot  of  the  harbor,  Fig.  122,  on  a scale  of  450  yards 
to  an  inch. 

Divide  the  length  of  the  base-line  AB,  which  is  equal  to 
1140  yards,  by  450,  and  the  quotient  2.53  will  express  the 
length  which  is  to  represent  the  base-line  on  the  paper. 

Draw  an  indefinite  line  AB,  to  represent  the  base;  and 
having  chosen  any  point,  as  A,  for  the  first  station,  lay  off  2.53 
inches  to  B.  The  other  extremity  of  the  base-line  will  thus  be 
determined. 

Then,  place  the  circular  protractor  at  A,  and  lay  off  the 
angle  BAB,  and  then  the  angle  EAG.  Next,  place  the  pro- 
tractor at  B,  and  lay  off  the  angles  ABE  and  EBC.  The 
intersection  of  the  lines  AE  and  BE  will  determine  the  station 
E.  Let  the  protractor  be  then  placed  at  this  point,  and  all 
the  angles  of  station  E laid  down. 

The  point  G,  where  EG  intersects  AG,  and  the  point  C, 
where  EC  intersects  BO,  will  then  be  found. 

By  placing  the  protractor  at  C and  G,  we  can  determine  the 
points  P and  F,  when  the  place,  on  the  paper,  of  all  the  stations 
will  be  known. 


SEC.  III.] 


PLOTTING  THE  TRIANGULATION. 


243 


To  unite  the  work  done  with  the  compass,  spread  the  com- 
pass-notes before  you,  and  draw  through  A a line  to  represent 
the  meridian.  The  course  AE  lies  to  the  west  of  this  meridian, 
and  makes  an  angle  of  12°  with  it. 

Then,  lay  oil  from  the  scale  the  distances  Aa , Ab , Aq , Ac, 
Ad,  Ae,  and  at  the  several  points  erect  perpendiculars  to  AE. 
Lay  off,  on  these  perpendiculars,  the  lengths  of  the  offsets,  and 
the  curve  traced  through  the  points  so  determined,  will  be  the 
margin  of  the  lake. 

At  E,  draw  a parallel  to  the  meridian  through  A,  and  lay 
down  the  course  EH,  which  is  easterly,  and  makes  an  angle 
of  50°  with  the  meridian.  Then,  lay  down  the  several  distances 
to  the  offsets,  and  draw  the  offsets  and  lay  off  their  lengths. 
Do  the  same  for  the  course  HI,  and  all  the  compass-work  will 
be  plotted. 

The  work  done  with  the  plane-table  is  united  to  the  work 
done  with  the  transit,  by  simply  reducing  it  to  the  same 
scale,  and  then  placing  the  line  AN  on  the  paper  of  the 
plane-table,  upon  the  line  AN,  drawn  on  the  plot  of  the 
triangulation. 

277.  Second  Method  of  Plotting. — Place  the  centre  of  the 
protractor  near  the  centre  of  the  paper,  and  draw  a line  through 
the  points  0 and  180°.  This  line  will  have  the  same  position  with 
the  circular  protractor  that  the  base-line  AB  had  with  the  limb  of 
the  transit. 

Then  lay  off,  from  the  0 point,  an  arc  equal  to  the  direction 
from  A to  E,  also  an  arc  equal  to  the  direction  A G,  and  through 
the  centre  point,  and  the  points  so  determined,  draw  lines. 
Lay  off  in  succession,  in  a similar  manner,  the  directions  taken 
at  all  the  stations ; and  through  the  centre  point,  and  the  points 
so  determined,  draw  lines,  and  designate  each  by  the  letters  of 
the  direction  to  which  it  corresponds. 


244 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI. 


Now,  since  all  the  lines  drawn  on  the  paper  have  the  same 
position  with  the  circular  protractor,  as  the  corresponding  lines 
on  the  ground  have  with  the  limb  of  the  transit,  it  follows 
that  each  direction  will  be  parallel  to  its  corresponding  line  upon 
the  ground. 

Hence,  any  line  may  be  drawn  parallel  to  that  passing  through 
0 and  180°,  to  represent  the  base-line  AB.  Having  drawn  such 
a line,  and  marked  a point  for  the  station  A,  lay  off  the  length 
of  the  base,  and  the  extremity  will  be  the  station  B. 

Through  A and  B,  so  determined,  draw  parallels  respectively 
to  the  lines  corresponding  to  the  directions  AE  and  BE,  and 
the  point  of  intersection  will  determine  station  E.  Through 
B and  E,  draw  parallels  to  the  lines  which  correspond  to  the 
directions  BC,  CE,  and  their  point  of  intersection  will  determine 
station  C.  Through  C and  E,  draw  lines  parallel  to  the  lines 
corresponding  to  the  directions  CE  and  ED,  and  the  point  of 
intersection  will  determine  D.  In  a similar  manner  we  may 
determine  the  stations  i^and  G. 

Method  of  Chords. 

278.  The  chord  of  a given  arc  is  equal  to  the  sine  of  half 
the  arc  with  double  the  radius. 

For,  let  DAF  be  any  given  angle, 
and  AH  a line  bisecting  it.  Let  DEC 
be  the  chord  of  the  arc  DC,  described 
with  a given  radius,  and  TIF  parallel 
to  CD,  the  sine  of  half  the  given 
angle,  to  a radius  AF  = 2 AC. 

Since  AF  = 2 AC,  we  have,  from  similar  triangles,  HF  = 
2 EC;  but  DC  =2  EC,  hence  HF=  CD. 

279.  To  lay  off  an  Angle. — To  avoid,  as  far  as  possible,  the 


SEC.  III.] 


PLOTTING  THE  TRIANGULATION. 


245 


use  of  fractions,  let  us  suppose  the  radius  of  the  table  of  natural 
sines  to  be  1 ten , or  10  inches. 

Take,  from  a scale,  5 equal  parts, 
with  which,  as  a radius,  from  the 
centre  A , describe  an  arc  CD.  Take 
from  the  table  the  natural  sine  of  A c 

Fig.  133. 

half  the  arc,  and  remove  the  decimal 

point  one  place  to  the  right;  the  result  will  express  the  sine  of 
half  the  arc  to  the  radius  10,  or  the  chord  of  the  arc  to  the 
radius  5.  From  the  same  scale,  take  this  sine  in  the  dividers, 
and  from  C , as  a centre,  describe  an  arc  cutting  CD  in  D ; draw 
AD,  and  CAD  will  be  the  angle  required. 


This  is  the  most  accurate  of  all  the  methods  of  laying  off  an 
angle,  and  it  may  also  be  applied  advantageously  to  the  second 
method  of  plotting,  thus : 


Draw  a fine  straight  line,  generally  in 
the  direction  of  the  meridian  or  of  the 
base-line  of  the  survey;  and  also  a line 
perpendicular  to  it.  From  the  point  of 
intersection,  as  a centre,  with  a radius  of 
5 equal  parts  of  the  scale,  describe  the  cir- 
cumference of  a circle  cutting  the  straight  lines  in  the  points 
marked  0 and  90°. 


To  lay  off  an  angle,  as,  for  instance,  the  angle  14°  29'.  The 
half  of  it  is  7°  14'  30",  the  natural  sine  of  which  is  0.12605, 
or  1.26  to  the  radius  of  10  inches.  Set  off  from  0 to  b , as  in 
the  figure,  this  distance  taken  from  the  scale,  and  through  the 
two  points  b , b,  thus  determined,  draw  a straight  line.  This 
line  should  pass  through  the  centre,  and  will  make  with  the 
line  (0,  0)  the  angle  14°  29' ; and  any  line  on  the  paper  drawn 
parallel  to  it,  will  make  with  the  line  (0,  0)  the  same  angle. 
The  further  application  is  obvious. 


BOOK  VII. 

LEVELING. 


SECTION  I. 

DEFINITIONS  AND  PRINCIPLES. 

280.  Leveling  is  the  art  of  determining  the  relative  dis- 
tances of  points  from  the  centre  of  the  earth. 

281.  A line  whose  points  are  all  equally  distant  from  the 
centre  of  the  earth,  is  called  a line  of  true  level ; and  a surface, 
all  whose  points  are  equally  distant  from  the  centre  of  the  earth, 
as  the  surface  of  still  water,  is  called  a level  surface. 

282.  One  point  is  said  to  be  above  another,  when  it  is  farther 
from  the  centre  of  the  earth  ; and  this  difference  of  distance 
from  the  centre,  is  called  the  difference  of  level  between  the  two 
points. 

283.  A straight  line  drawn  tangent 
to  a line  of  true  level,  at  any  point,  is 
a horizontal  line,  and  is  called  the  line 
of  apparent  level.  Thus,  Fig.  135,  if  G 
is  the  centre  of  the  earth,  and  AEF aline 
of  true  level,  A BD  is  the  line  of  apparent 
level.  This  is  the  line  of  level  determined 
by  an  instrument.  The  difference  be- 
tween the  apparent  and  true  level  of  the  points  A and  E,  is  BE, 
the  excess  of  the  secant  GB,  of  the  arc  AE,  over  the  radius  CE. 


c 

Fig.  135. 


SEC.  I.] 


DEFINITIONS. 


247 


284.  To  find  a general  formula  for  computing  this  excess,  we 
have  (Geom.,  B.  IV,  Prop.  XXX), 

Xfi2  = BE  (BE  + 2EC); 


but,  since  the  arc  AE  is  very  small  in  comparison  with  the 
radius  of  the  earth,  the  arc  AE  will  not  differ  sensibly  from 
the  tangent  AB ; the  diameter  2 EC  may,  for  the  same  reason, 
be  taken  for  the  secant  (BE+%EC)\  hence, 


AE~  — BEx2EC,  or,  dividing  by  2 EC, 


BE  = 


AE 4 
2EO  * 


(!)• 


If  we  take  the  mean  diameter  of  the  earth  to  be  7919  miles, 
formula  (1)  gives 


BE  = 


AE 2 
7919 


(2),  hence, 


The  departure  of  the  apparent  from  the  true  level , starting 
from  a given  point , is  equal  to  the  square  of  the  distance  to  the 
second  point,  divided  hy  the  diameter  of  the  earth. 

If,  in  formula  (2),  we  give  to  AE,  in  succession,  every  value 
from  1 chain  to  any  given  number  of  chains  (say  100),  and 
reduce,  at  the  same  time,  both  terms  of  the  fraction  to  inches, 
a table  may  be  computed  as  on  next  page. 

Observe,  that  when  the  distance  AE  = 80  chains  = 1 mile, 
that  BE  is  = 8.001  inches,  or  two-thirds  of  a foot,  very  nearly  ; 
and  for  any  other  distance,  d,  in  miles,  we  have, 


1 2 : d2  : : § of  a foot  : § d2 ; 

hence,  we  have  the  following  easy  rule  for  finding  the  correc- 
tion of  curvature  in  feet : 

The  correction  for  curvature,  in  feet,  is  equal  to  two-thirds 
of  the  square  of  the  distance  in  miles. 


248 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


Table  showing  the  differences,  in  inches,  between  the  true  and 
apparent  level,  for  distances  between  1 and  100  chains. 


Chains. 

Inches. 

Chains. 

Inches. 

Chains. 

Inches. 

Chains. 

Inches. 

1 

.001 

26 

.845 

51 

3.255 

76 

7.221 

2 

.005 

27 

.911 

52 

3.380 

77 

7.412 

3 

.011 

28 

.981 

53 

3.511 

78 

7.605 

4 

.020 

29 

1.051 

54 

3.645 

79 

7.802 

5 

.031 

30 

1.125 

55 

3.781 

80 

8.001 

6 

.045 

31 

1.201 

56 

3.925 

81 

8.202 

7 

.061 

32 

1.280 

57 

4.061 

82 

8.406 

8 

.080 

33 

1.360 

58 

4.205 

83 

8.612 

9 

.101 

34 

1.446 

59 

4.351 

84 

8.832 

10 

.125 

35 

1.531 

60 

4.500 

85 

9.042 

11 

.151 

36 

1.620 

61 

4.654 

86 

9.246 

12 

.180 

37 

1.711 

62 

4.805 

87 

9.462 

13 

.211 

38 

1.805 

63 

4.968 

88 

9.681 

14 

.245 

39 

1.901 

64 

5.120 

89 

9.902 

15 

.281 

40 

2.003 

65 

5.281 

90 

10.126 

16 

.320 

41 

2.101 

66 

5 443 

91 

10.351 

17 

.361 

42 

<Nt 

O 

OD 

67 

5.612 

92 

10.587 

18 

.405 

43 

2.311 

68 

CO 

93 

10.812 

19 

.451 

44 

2.240 

69 

5.955 

94 

11.046 

20 

.500 

45 

2.531 

70 

6.125 

95 

11.233 

21 

.552 

46 

2.646 

71 

6.302 

96 

11.521 

22 

.605 

47 

2.761 

72 

6.480 

97 

11.763 

23 

.661 

48 

2.880 

73 

6.662 

98 

12.017 

24 

.720 

49 

3.004 

74 

6 .'846 

99 

12.246 

25 

.781 

50 

3.125 

75 

7.032 

100 

12.502 

SEC.  II.] 


INSTRUMENTS. 


249 


SECTION  II. 

I N STRU  M ENTS. 

285.  The  Y Level. — A Level  is  an  instrument  used  to 
indicate  a horizontal  line,  and  also,  to  determine  the  difference 
of  level  of  any  two  points  on  the  surface  of  the  earth. 


250 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


The  part  of  the  instrument  shown  in  Fig.  13G,  rests  on  a 
tripod,  to  which  it  is  made  fast,  and  on  which  it  is  leveled  by 
means  of  two  leveling  plates,  such  as  are  described  in  the  account 
of  the  Transit,  Art.  177. 

The  telescope  rests  on  vertical  supports  called,  from  their 
shape,  Y’s  or  Wyes,  and  is  confined  in  the  wyes  by  loops  or 
clips,  which  are  fastened  by  tapering-pins. 

The  telescope  has  at  each  end  a ring  of  bell-metal,  turned 
very  truly  and  both  rings  of  exactly  the  same  diameter ; by  these 
it  revolves  in  the  wyes,  or  can  be  at  pleasure  clamped  in  any  posi- 
tion when  the  clips  of  the  wyes  are  brought  down  upon  the  rings, 
by  pushing  in  the  tapering-pins.  It  has  a rack  and  pinion 
movement  to  both  object-glass  and  eye-piece. 

A spirit-level  or  ground  bubble-tube  is  attached  to  the  under 
side  of  the  telescope,  and  furnished  at  the  different  ends  with 
screws  for  movement  in  both  horizontal  and  vertical  directions. 

The  level  scale  which  extends  over  the  whole  length  is  gradu- 
ated to  tenths  of  an  inch,  and  figured  at  every  fifth  division, 
counting  from  zero  at  the  centre  of  the  aperture  of  the  tube  through 
which  the  glass  vial  appears  ; the  scale  is  set  close  to  the  glass. 

The  wyes  are  perpendicular  to  a level-bar,  to  which  they  are 
screwed  fast ; each  wye  has  two  nuts,  both  adjustable  with  the 
ordinary  steel  pin.  Connected  with  the  level-bar  is  the  head  of 
the  tripod-socket. 

The  tripod-socket  is  compound ; the  interior  spindle  D, 
Fig.  137,  upon  which  the  whole  instrument  is  supported,  is 
made  of  steel,  and  nicely  ground,  so  as  to  turn  evenly  and 
firmly  in  a hollow  cylinder  of  bell-metal;  this  again  has  its 
exterior  surface  fitted  and  ground  to  the  main  socket  EE  of  the 
tripod-head. 

The  bronze  cylinder  is  held  upon  the  spindle  by  a washer  and 
screw,  the  head  of  the  last  having  a hole  in  its  centre,  through 
which  the  string  of  the  plumb-bob  is  passed. 


SEC.  II.] 


INSTRUMENTS. 


251 


The  upper  part  of  the  instrument,  with  the  socket,  may  thus 
be  detached  from  the  tripod-head ; and  tills  also  can  be  un- 


screwed from  the  legs,  so  that  both  may  bo  conveniently 
packed  in  the  box. 

A little  under  the  upper  parallel  plate  of  the  tripod-head,  and 
in  the  main  socket,  is  a screw  which  can  be  moved  into  a corres 


252 


ELEMENTS  OF  SURVEYING. 


[ROOK  VII. 


ponding  groove,  turned  on  the  outside  of  the  hollow  cylinder, 
and  thus  made  to  hold  the  instrument  in  the  tripod  when  it  is 
carried  upon  the  shoulders. 

Before  using  the  Level,  it  must  be  adjusted.  The  adjust- 
ment consists  in  bringing  the  different  parts  to  their  proper 
places. 

The  line  of  collimation  is  the  axis  of  the  telescope.  With 
this  axis,  the  line  drawn  through  the  centre  of  the  eye-glass 
and  the  intersection  of  the  cross-wires,  within  the  barrel  of 
the  telescope,  ought  to  coincide. 


FIRST  ADJUSTMENT. 

To  fix  the  intersection  of  the  cross-wires  in  the  axis  of  the 

telescope. 

286.  Having  screwed  the  tripod  to  the  instrument,  extend 
the  legs  and  place  them  firmly.  Then  loosen  the  clamp-screw 
and  direct  the  telescope  to  a small,  well-defined,  and  distant 
object.  Then  slide  the  eye-glass  till  the  cross-wires  are  seen 
distinctly ; after  which  adjust  the  object-glass  to  its  proper 
focus,  when  the  object  and  the  cross-wires  will  be  distinctly 
seen.  Note  now  the  precise  point  covered  by  the  intersection 
of  the  cross-wires. 

Having  done  this,  revolve  the  telescope  in  the  Y’s  half 
round,  when  the  attached  level  will  come  to  the  upper  side. 
See  if,  in  this  position,  the  horizontal  wire  appears  above  or 
below  the  point,  and  in  either  case,  loosen  the  one  and  tighten 
the  other  of  the  two  screws  which  work  the  horizontal  wire, 
until  it  has  been  carried  over  half  the  space  between  its  last 
position  and  the  observed  point ; bring  it  the  rest  of  the  way 
by  the  leveling  screws.  Carry  the  telescope  back  to  its  place  ; 
direct  again  the  intersection  of  the  cross-wires  to  the  point,  and 
repeat  the  operation  till  the  horizontal  wire  neither  ascends 


SEC.  II.] 


INSTRUMENTS. 


253 


nor  descends  while  the  telescope  is  revolved.  A similar  process 
will  arrange  the  vertical  wire,  and  the  line  of  collimation  is  then 
adjusted. 

SECOND  ADJUSTMENT. 

To  make  the  axis  of  the  attached  level  parallel  to  the  line  of 

collimation. 

287.  Turn  the  leveling-screws  until  the  bubble  of  the  level 
stands  at  the  middle  of  the  tube.  Then  open  the  loops  and 
reverse  the  telescope.  If  the  bubble  still  stands  at  the  middle 
of  the  tube,  the  axis  of  the  level  is  horizontal ; but  if  not,  it  is 
inclined,  the  bubble  being  at  the  elevated  end.  In  such  case, 
raise  the  depressed  or  depress  the  elevated  end  by  means  of  the 
small  screw  provided  for  the  purpose,  half  the  inclination  ; and 
then  with  the  leveling  screws  bring  the  level  to  a horizontal 
position.  Reverse  the  telescope  in  the  Y’s  and  make  similar 
corrections  again  ; and  proceed  thus  until  the  bubble  stands  in 
the  middle  of  the  tube,  in  both  positions  of  the  telescope  ; the 
axis  of  the  level  is  then  horizontal. 

Let  the  telescope  be  now  revolved  in  the  Y’s.  If  the  bubble 
continues  in  the  middle  of  the  tube,  the  axis  of  the  level  is  not 
only  horizontal,  but  also  parallel  to  the  line  of  collimation.  If, 
however,  the  bubble  recedes  from  the  centre,  the  axis  of  the 
level  is  inclined  to  the  line  of  collimation,  and  must  be  made 
parallel  to  it  by  means  of  two  small  screws,  which  work  hori- 
zontally. By  loosening  one  of  them  and  tightening  the  other, 
the  level  is  soon  brought  parallel  to  the  line  of  collimation  ; 
and  then,  if  the  telescope  be  revolved  in  the  Y’s,  the  bubble  will 
continue  at  the  middle  of  the  point  of  the  tube.  It  is,  however, 
difficult  to  make  the  first  part  of  this  adjustment,  while  the  axis 
of  the  level  is  considerably  inclined  to  the  line  of  collimation ; 
for,  even  if  the  level  be  truly  horizontal  in  one  position  of  the 
telescope,  after  it  is  reversed  there  will  be  but  one  corresponding 


254 


ELEMENTS  OE  SURVEYING. 


[ROOK  VII. 


position  in  which  the  bubble  will  stand  at  the  middle  of  the 
tube.  This  suggests  the  necessity  of  making  the  first  part  of 
the  adjustment  with  tolerable  accuracy;  then,  having  made  the 
second  with  care,  re-examine  the  first,  and  proceed  thus  till  the 
adjustment  is  completed. 


THIRD  ADJUSTMENT. 

To  make  the  level  and  the  line  of  collimation  perpendicular  to 
the  axis  of  the  instrument,  or  parallel  to  the  level-bar. 

288.  Loosen  the  clamp-screw  and  turn  the  bar  until  the 
level  comes  directly  over  two  of  the  leveling  screws.  By  means 
of  these  screws,  make  the  level  truly  horizontal.  Then,  turn 
the  level  180°  upon  its  vertical  axis  ; if,  during  the  revolution,  it 
continue  horizontal,  it  must  be  at  right  angles  to  the  axis  of 
the  instrument  about  which  it  has  been  revolved.  But  if,  after 
the  revolution,  the  level  be  not  horizontal,  rectify  half  the 
error  with  the  screws  at  M and  R,  Fig.  137,  and  half  with  the 
leveling  screws.  Then  place  the  bar  over  the  other  two  leveling 
screws,  and  make  the  same  examinations  and  corrections  as 
before ; and  proceed  thus,  until  the  level  can  be  turned  entirely 
around  without  displacing  the  bubble  at  the  centre.  When  this 
can  be  done,  it  is  obvious  that  the  level  and  the  line  of 
collimation  are  at  right  angles  to  the  axis  of  the  instrument 
about  which  they  revolve ; and  since  the  axis  is  carefully  ad- 
justed by  the  maker,  at  right  angles  to  the  bar,  it  follows 
that  the  line  of  collimation,  the  level,  and  the  bar,  are  parallel 
to  each  other. 

rt  is  always  necessary  to  examine  the  adjustments  frequently, 
in  order  to  secure  satisfactory  results. 


SEC.  II.] 


INSTRUMENTS. 


255 


LEVELING  RODS. 


289.  The  leveling  rods  are  used  to  determine  the  points  at 
which  a given  horizontal  line  intersects  lines 

that  are  perpendicular  to  the  surface  of  the 
earth,  and  to  show  the  distances  of  such  points 
of  intersection  from  the  ground. 

There  are  three  kinds  of  rods  used  by  En- 
gineers, known  as  the  New  York,  Philadelphia, 
and  Boston  or  Yankee  rods.  The  Philadelphia 
Rod  is  divided  to  tenths,  and  reads  to  two-hun- 
dredths of  a foot.  The  New  York  and  Boston 
rods  are  divided  to  hundredths  of  a foot,  and 
read  by  verniers  to  thousandths.  They  are  all 
sliding  rods. 

290.  New  York  Rod. — This  rod,  which  is 
shown  in  Fig.  138,  is  cut  in  two  parts  so  that 
both  ends  may  be  exhibited.  It  is  made  of 
maple  or  satin-wood,  in  two  pieces,  sliding  one 
from  the  other,  always  in  the  same  direction,  so 
that  the  same  end  is  always  held  on  the  ground, 
and  the  graduations  start  from  that  point. 

The  graduations  are  made  to  tenths  and  hun- 
dredths of  a foot,  the  tenth  figures  being  black, 
and  the  feet  marked  with  a large  red  figure. 

A target  is  used  to  indicate  where  the  hori- 
zontal line  cuts  the  rod. 

The  face  of  the  target  is  divided  into  quad- 
rants, by  a horizontal  and  a vertical  diameter; 
and  these  diameters  are  the  boundaries  of  alter- 
nate colors  with  which  the  diagonal  quadrants 
are  painted. 

The  opening  in  the  face  of  the  target  is  a fig.  m. 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


256 

little  more  than  a tenth  of  a foot  long,  so  that  in  any  position  a 
tenth,  or  a foot  figure,  can  be  seen  on  the  surface  of  the  rod. 

The  right  edge  of  the  opening  is  chamfered,  and  divided  into 
ten  equal  spaces,  corresponding  with  nine-hundredths  on  the 
rod  ; the  divisions  start  from  the  horizontal  line  which  separates 
the  colors  of  the  face.  The  vernier  reads  to  thousandths  of  a 
foot. 

For  heights  less  than  six  and  a half  feet,  the  target  is  moved 
along  the  sliding  part,  to  which  it  is  slightly  attached  by  springs, 
and  to  which  it  may  be  permanently  attached  by  a clamp-screw, 
and  the  reading  is  made  by  the  vernier  on  the  target. 

When  a greater  height  is  required,  the  horizontal  line  of  the 
target  is  fixed  at  that  point,  and  the  upper  half  of  the  rod,  car- 
rying the  target,  is  moved  out  of  the  lower,  the  reading  being 
now  obtained  by  a vernier  on  the  graduated  side,  up  to  an  eleva- 
tion of  twelve  feet. 

TESTS  OF  ADJUSTMENT. 

291.  There  is  a method  of  testing  the  adjustments  of  the 
Y level,  which  ought  not  to  be  neglected,  since  all  the  results 
depend  on  the  accuracy  of  the  instrument.  The  method  is 
this: 


The  level  being  adjusted,  place  it  at  any  convenient  point, 
as  0 (Fig.  139).  At  equal  distances  of  about  300  feet  in  opposite 


SEC.  II.] 


INSTRUMENTS. 


257 


directions  from  the  instrument,  drive  two  pegs  firmly  into  the 
ground  and  take  readings  of  the  rod  upon  them.  The  difference 
of  the  readings  will  be  the  difference  of  level  of  the  tops  of 
the  pegs,  even  though  the  level  be  out  of  adjustment.  Now  set 
the  level  at  about  50  feet  beyond  either  peg,  nearly  in  line  with 
them,  and  again  take  rod-readings  upon  them ; the  difference  of 
these  new  readings,  corrected  for  curvature  of  the  earth , should 
equal  the  difference  of  level  of  the  pegs  as  found  before.  If  the 
readings  are  not  what  they  should  be,  the  adjustment  may  be 
perfected  thus:  Call  the  further  peg  a,  the  nearer  peg  b , 


and  the  position  of  the  instrument  c.  (Fig.  140).  Let  R be  the 
first  reading  at  a , and  R'  that  at  b ; let  r be  the  second  reading 
at  a,  corrected  for  curvature,  and  r'  the  corresponding  reading 
at  5,  also  corrected  ; let 

(R-R')-(r-r')  = ±D; 

then  we  have 

ab  (=  600  ft.)  : ac  (=  650  ft.)  : : ± D : ± d. 

Add  the  correction  d to  r,  and  having  set  the  target  to  this 
reading  on  a,  bring  the  horizontal  wire  to  coincide  with  it  by 
the  adjusting  screws.  See  also  that  the  bubble  is  in  the  middle 
of  the  run  at  the  same  time. 


258 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


SECTION  III. 

LEVE  LI  NG  IN  THE  FIELD. 

292.  The  operations  of  leveling  may  be  undertaken . 

1st.  For  the  purpose  of  determining  the  difference  of  level 
between  two  given  points; 

2d.  For  the  purpose  of  obtaining  a section  or  profile  along 
a given  line,  as  in  the  preliminary  surveys  for  railroads  and 
canals ; 

And,  as  will  be  described  hereafter, 

3d.  For  the  purpose  of  determining  the  contour  lines  in  a 
topographical  survey  ; 

4thly.  For  the  purpose  of  determining  the  volume  of  any 
given  mass  of  earthwork  or  masonry ; as  the  measurement 
of  excavations  and  embankments  for  canals  and  railroads ; 
and, 

5thly.  For  the  purpose  of  determining  and  indicating 
boundaries  for  filling  and  excavation  ; such  as  setting  slope 
stakes,  &c. 


Difference  of  Level  between  Two  Points. 

293.  When  it  is  proposed  to  find  the  difference  of  level  of 
any  two  objects,  or  stations,  all  levels  made  in  the  direction  of 
the  station  at  which  the  work  is  begun,  are  called,  for  the  sake 
of  distinction  merely,  back-sights  ; and  levels  taken  in  the  direc- 
tion of  the  other  station,  foresights. 

Before  going  on  the  field  with  the  level,  rule  three  columns, 
as  below,  and  head  them,  stations,  back-sights,  fore-sights. 


SEC.  III.] 


LEVELING  IN  THE  FIELD. 


259 


FIELD  NOTES. 


Stations. 

+ Back-Sights. 

— Fore-Sights. 

1 

10 

3 

2 

11.6 

0 

3 

6.8 

4.9 

4 

3.9 

8.3 

Sums  ....  32.3 

16.2 

Dif.  of  level  . . 16.1 

16.2 

Find  the  difference  of  level  between  any  two  points,  as  A and 

G-  (Fig.  141). 

294.  The  level  being  adjusted,  place  it  at  any  point,  as  B,  as 
nearly  in  the  line  joining  A and  G , as  may  be  convenient. 


Place  a leveling  rod  at  A.  Make  the  level  horizontal  by  means 
of  the  leveling  screws;  turn  the  telescope  to  the  rod  at  A,  and 
direct  the  rodman  to  raise  the  target  until  the  horizontal  lino 


260 


ELEMENTS  OF  SURVEYING. 


[HOOK  VJI. 


ab  pierces  its  centre;  then  note  the  distance  Ab  (equal  to  10 
feet  in  the  present  example)  and  enter  it  in  the  column  of 
back-sights  opposite  station  1. 

Send  the  rodman  forward  to  some  point,  as  N,  in  the  pro- 
posed direction,  and  sight  to  the  rod  as  before;  enter  the  dis- 
tance Na,  equal  to  3 feet,  in  the  column  of  fore-sights  opposite 
station  1 (B).  Then  remove  the  level  to  a convenient  point, 
as  G (2).  Direct  the  rodman  to  run  up  the  vane  to  the  proper 
height;  then  make  the  back-sight,  and  enter  it,  Nd  = 11.6  feet, 
in  the  column  of  back-sights,  opposite  station  2.  Let  the  rod- 
man  then  be  sent  forward  to  a convenient  point,  as  M,  and 
make  the  fore-sight  to  /;  and  enter  Mf  = 0,  in  the  column  of 
fore-sights,  opposite  station  2 ( C ).  Remove  the  level,  in  succes- 
sion, to  D and  E,  and  make  similar  levels  at  those  points,  and 
enter  the  results  in  the  column  of  back-sights  and  fore-sights, 
opposite  station  3 (D),  and  4 ( E ). 

It  is  evident  from  the  figure,  that  the  difference  of  level  NF, 
between  A and  N,  is  equal  to  the  back-sight  bA,  diminished  by 
the  fore-sight  aN ; also  that  the  difference  of  level  between 
N and  M,  is  equal  to  the  back-sight  dN,  diminished  by  the 
fore-sight  0,  and  since  each  set  of  observations  is  entirely  inde- 
pendent of  every  other  set,  we  may  infer  that  the  difference  of 
level  between  two  consecutive  points,  as  determined  by  the  same 
position  of  the  level,  is  equal  to  the  back-sight,  diminished  by  the 
fore-sight.  If  the  fore-sight  is  greater  than  the  back-sight,  the 
difference  will  be  affected  with  a minus  sign,  a result  which 
shows  that  the  second  point  is  lower  than  the  first ; and, 

Generally,  the  difference  of  level  between  any  two  points,  deter- 
mined as  above,  is  equal  to  the  sum  of  the  back-sights  diminished 
by  the  sum  of  the  foresights.  If  the  result  is  plus,  the  second 
point  is  higher  than  the  first;  if  negative,  it  is  lower. 

in  the  example  given,  the  difference  of  level  between  A and 
G,  is  16  feet  and  1 tenth. 


SEC.  III.  | 


LEVELING  IN  THE  FIELD. 


261 


295.  In  the  above  example,  we  did  not  regard  the  difference 
between  the  true  and  apparent  level.  If  it  is  necessary  to 
ascertain  the  result  with  extreme  accuracy,  this  difference  must 
be  considered  ; and  then,  the  horizontal  distances  between  the 
level,  at  each  of  its  positions,  and  the  rods,  must  be  measured, 
and  the  apparent  levels  diminished  by  the  differences  of  level ; 
which  differences  can  be  found  from  the  table. 


EXAM  p l e . 


Stat. 

Back-sts. 

Distances. 

Fore-st. 

Distances. 

Cor.  back-sts. 

Cor.  for-sts. 

1 

9.8 

20  ch. 

1.6 

32  ch. 

9.7583 

1.4933 

2 

8.7 

25  ch. 

2.4 

28  ch. 

8.6349 

2.3183 

3 

5.2 

18  ch. 

3.1 

16  ch. 

5.1663 

3.0734 

4 

10.3 

29  ch. 

1.9 

87  ch. 

10.2124 

1.1115 

5 

11.0 

45  ch. 

2.5 

72  ch. 

10.7891 

1.9600 

44.5610 

8.9565 

In  this  example,  the  first  column  shows  the  stations ; the 
second,  the  back-sights ; the  third,  the  distances  from  the  level 
in  each  of  its  positions  to  the  back  rod ; the  fourth,  the  fore- 
sights ; the  fifth,  the  distances  from  the  level  to  the  forward  rod; 
the  sixth  and  seventh,  are  the  columns  of  back  and  fore-sights, 
corrected  by  the  difference  of  level.  The  corrections  are  thus 
made  : The  difference  of  level  in  the  table  corresponding  to  20 
chains,  is  5 tenths  of  an  inch,  or  .0417  of  a foot  nearly;  which 
being  subtracted  from  9.8  feet,  leaves  9.7583  feet  for  the  corrected 
back-sights;  this  is  entered  opposite  station  1 in  the  sixth 
column.  The  difference  of  level  corresponding  to  32  chains,  is 
1.280  inches,  or  .1067  feet,  nearly ; which  being  subtracted  from 
the  apparent  level,  1 foot  6 tenths,  leaves  1.4933  feet,  for  the  true 
fore-sight  from  station  1.  The  other  corrections  are  made  in 
the  same  manner. 


262 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII, 


The  sum  of  the  back-sights  being  44.5610  feet,  and  the 
sum  of  the  fore-sights  8.9565  feet,  it  follows  that  the  difference, 
35.6045  feet,  is  the  true  difference  of  level. 

296.  In  finding  the  true  from  the  apparent  level,  we  have 
not  regarded  the  effect  caused  by  refraction  on  the  apparent 
elevation  of  objects,  as  well  because  the  refraction  is  different 
in  different  states  of  the  atmosphere,  as  because  the  corrections 
are  for  short  distances  inconsiderable.  The  error  occasioned  by 
refraction  is  opposite  to,  and  tends  to  diminish,  the  error  occa- 
sioned by  the  curvature  of  the  earth.  If  desired,  it  may  be 
corrected  by  diminishing  the  effect  of  the  earth’s  curvature  by 
one-seventh  of  itself. 

297.  The  small  errors  that  would  arise  in  regarding  the 
apparent  as  the  true  level,  may  be  avoided  by  placing  the  leveling 
rods  at  equal  distances  from  the  level.  In  such  case,  it  is  plain, 
1st,  that  equal  corrections  must  be  made  in  the  fore  and  back- 
sights ; and,  2dly,  that  when  the  fore  and  back-sights  are 
diminished  equally,  the  result,  which  is  always  the  difference  of 
their  sums,  will  not  be  affected. 

This  method  should  always  be  followed,  if  practicable,  as  it 
avoids  the  trouble  of  making  corrections  for  the  difference  of  true 
and  apparent  level. 

The  differences  between  the  true  and  apparent  level,  being 
very  inconsiderable  for  short  distances,  if  only  ordinary  accuracy 
is  required,  it  will  be  unnecessary  to  make  measurements  at  all, 
Care,  however,  ought  to  be  taken,  in  placing  the  leveling  rods, 
to  have  them  as  nearly  equidistant  from  the  level,  which  can  be 
effected  by  the  rodman,  by  pacing  the  distance  from  back-sight 
rod  to  level,  and  from  level  to  fore-sight  rod  ; and  if  the  distances 
are  unequal,  let  the  next  distances  also  be  made  unequal ; that  is, 
if  the  back-sight  is  the  longer  in  the  first  case,  let  it  be  made 
proportionably  shorter  in  the  second,  and  the  reverse. 


SEC.  IV.] 


SECTION  LEVELING. 


263 


SECTION  IV. 

SECTION  LEVELING. 

298.  In  the  surveys  which  precede  the  construction  of  roads, 
railroads,  canals,  dikes,  or  other  similar  earthworks,  the  surveyor 
must  make  such  measurements  as  are  necessary  to  enable  him 
to  estimate  the  volume  of  the  material  to  be  removed.  In 
addition,  therefore,  to  the  horizontal  measurements  made  in 
connection  with  the  location  of  the  work,  vertical  dimensions, 
or  heights,  are  also  necessary,  and  are  taken  at  every  important 
change  in  the  inclination  of  the  surface  along  the  line  of  the 
survey. 

These  heights  are  taken  by  the  level  and  rod,  and  are 
simply  vertical  distances  of  points  along  the  surface  above  an 
assumed  level  line  called  the  datum  line. 

299.  In  the  survey  of  a long  line  of  railway  or  canal,  one 
of  whose  termini  is  in  the  vicinity  of  tide-water,  the  datum  line 
is  usually  assumed  at  the  level  of  mean  high-water.  In  cases 
of  surveys  entirely  inland,  the  datum  line  is  taken  at  some 
convenient  depth  below  the  beginning  point  of  the  survey,  and 
at  such  a distance  that  it  shall  be  below  the  entire  line  on  the 
surface.  For  such  surveys,  the  system  of  notes  described  in 
the  preceding  section  is  insufficient. 

300.  As  the  survey  progresses,  fixed  points  of  reference, 
called  benches , are  located  in  the  vicinity  of  the  line.  Permanent 
objects  are  usually  selected  for  benches  ; such  as  rocks,  build- 
ings, or  trees,  and  at  such  distances  from  the  line  of  the  work 
as  to  be  undisturbed  by  the  subsequent  construction. 


264 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


301.  Temporary  benches,  employed  merely  while  changing 
the  position  of  the  leveling  instrument,  are  called  turning  'points. 
In  either  case,  a well-defined  point  must  be  provided — one  not 
easily  disturbed  by  a blow,  and,  moreover,  one  upon  which  the 
rod  can  be  held  vertically. 

302.  In  order  to  understand  the  field  operations  and  the 
mode  of  keeping  notes,  it  will  be  necessary  to  comprehend  the 
principle  involved  in  all  leveling  practice. 

Suppose  that  the  depths  of  different  points  of  the  bottom  of  a 
shallow  lake  are  required  ; these  could  be  readily  obtained  by 
measuring  the  distances  from  the  surface  of  the  water  to  the 
bottom  by  means  of  a sounding  line,  or  in  winter  by  cutting 
holes  in  the  ice  and  measuring  the  distances  to  the  bottom  with 
a pole. 

In  this  illustration  the  various  points  on  the  bottom  are 
located  with  reference  to  the  plane  of  the  surface  of  the 
water. 

The  practice  of  leveling  is  identical  in  principle.  The  hori- 
zontal surface  of  reference,  from  which  to  deduce  relative  heights, 
is  generated  by  the  line  of  collimation  as  the  telescope  is  revolved 
about  the  vertical  axis,  and  the  “soundings”  to  points  below 
this  plane  of  reference  are  made  with  a leveling  rod,  whose 
lower  end  rests  upon  the  point  to  be  located,  and  whose  target 
is  moved  so  that  the  plane  of  reference  shall  cut  its  horizontal 
line. 

When  the  instrument  is  set  up  at  a new  station,  the  distance 
of  its  new  plane  of  reference  above  or  below  the  previous 
one,  must  be  determined  in  order  to  secure  continuity  of  the 
work. 

303.  The  following  example  will  exhibit  the  method  of  re- 
cording llio  notes  of  a section  level.  The  datum  line  is  assumed 
to  bo  thirty  feet  below  the  first  bench.  When  the  field-book 


sec.  iv.J 


SECTION  LEVELING. 


265 


is  of  the  ordinary  pocket  size,  the  seven  columns  of  notes  will 
generally  occupy  two  opposite  pages ; the  first  five  being  upon 
the  left-hand  page. 


Fig.  142. 


Dist. 

+ Sight. 

Ht.  of  Ins. 

— Sight. 

Surface 

Height. 

Grade 

Height. 

Remarks. 

Bench. 

1.637 

31.637 

30. 

Bench  on  top 
of  fence-post 

0 

2.1 

29.5 

30  ft.  north  of 
0 stake. 

1 

1.8 

29.8 

^60 

0.9 

30.7 

2 

3.4 

28.2 

3 

10.8 

20.8 

T.  P. 

1.910 

22.134 

11.413 

20.224 

4 

5.8 

16.3 

5 

9.0 

13.1 

550 

10.4 

11.7 

6 

9.8 

12.3 

7 

10.6 

11.5 

The  bench  having  been  selected  and  marked,  its  location  is 
described  in  the  column  of  remarks. 

The  level  is  adjusted  in  some  convenient  place  in  the  vicinity, 


266 


ELEMENTS  OF  SURVEYING. 


[BOOK  VI r. 


and  the  reading  of  the  rod  is  taken  upon  the  bench.  In  the 
above  example  it  is  1.637.  As  the  bench  is  30  feet  above  the 
assumed  datum  line,  the  height  of  the  instrument  (or  line  of 
colli matioti)  above  this  datum  line  is  31.637  feet. 

The  reading  is  recorded  against  Bench , in  the  column  of 
+ sights , and  the  “height  of  instrument”  is  recorded  in  its 
proper  column,  in  the  same  line. 

By  referring  to  the  above  diagram  it  will  be  readily  seen, 
that  to  obtain  the  height  of  the  different  points  0,  1,  l60,  &c., 
above  the  datum  line,  it  is  only  necessary  to  take  the  readings 
of  the  rod,  at  these  stations,  and  subtract  them  from  31.637. 
Such  readings,  therefore,  are  appropriately  termed  minus  sights , 
and  are  recorded  in  the  4th  column.  As  these  readings  are 
taken  only  to  the  nearest  tenth  of  a foot,  they  are  taken  much 
more  rapidly  than  the  bench  readings.  The  subtractions  by 
which  the  surface  heights  are  found,  may  be  worked  in  the 
field  or  not,  as  the  surveyor  chooses.  The  unit  of  measurement, 
in  the  column  of  distances,  is  usually  the  engineer’s  chain  of 
100  feet.  Readings  are  taken  at  intermediate  points  (as  at  160 
feet  in  the  above  example)  when  there  are  abrupt  changes  in 
the  inclination  of  the  surface. 

306.  When  it  becomes  necessary  to  change  the  position  of 
the  level,  such  measures  must  be  taken  as  will  insure  the  exact 
“height  of  instrument,”  in  the  new  position. 

To  effect  this,  a carefully-selected  hard  point  is  found  (not 
necessarily  on  the  exact  line  of  the  survey,  but  as  far  forward 
as  convenience  and  accuracy  will  permit),  and  a reading  of  the 
rod  is  taken  upon  it,  to  thousandths. 

If  likely  to  be  used  for  a single  occasion  only,  it  is  called  a 
“ turning-point ,”  and  marked  T.  P.  in  the  distance  column; 
otherwise  it  is  called  a Bench , and  its  location  is  described  in 
the  column  of  remarks. 


SEC.  1 V.] 


SECTION  LEVELING. 


267 


A turning-point  is  taken  between  stations  3 and  4,  in  the 
above  example.  The  reading  of  the  rod,  upon  it,  is  11.413. 
This  is  recorded  in  the  — sight  column,  and  the  surface  height 
of  the  point  is  at  once  found  as  before,  and  recorded  in  the 
column  of  “ Heights.  ” 

The  level  is  next  carried  forward  to  a new  position,  adjusted, 
and  directed  again  upon  the  rod  still  held  upon  the  turning- 
point.  The  reading  is  taken  to  thousandths.  This,  when  added 
to  the  height  of  the  turning-point,  evidently  gives  the  height  of 
instrument  in  its  new  position.  It  is  recorded,  therefore,  as  a + 
sight.  The  survey  is  now  continued  by  taking  — sights  at  the 
various  points  along  the  line  until  it  becomes  again  necessary  to 
change  the  position  of  the  level. 

In  the  above  example,  the  reading  of  the  rod  upon  the 
turning-point,  from  the  second  position  of  the  level,  is  1.910. 
The  height  of  the  point  upon  which  the  rod  stands  is  20.224. 
The  sum  of  these,  or  22.134,  is  the  “Height  of  Inst.”  for  the 
second  set  of  — sights.  The  successive  subtractions  of  the 
readings  from  the  Height  of  Instrument,  give  the  surface  heights 
as  before. 

The  most  extended  section  levels  are  but  repetitions  of  this 
process. 

305.  The  rules  for  taking  and  recording  field-notes  in  section 
leveling  are  as  follows: 

I.  Thr  “ distances  ” recorded  in  the  first  column  are  the 
horizontal  measurements,  in  chains,  from  the  beginning 
of  the  survey  to  the  points  ivhose  heights  are  to  be  deter- 
mined. The  heights  are  talcen  at  each  whole  chain,  and 
at  such  intermediate  points  as  the  irregularities  of  the 
surface  require. 

II.  The  first  reading  of  the  rod,  after  each  setting  of 


ELEMENTS  OF  SURVEYING. 


268 


[book  VII 


the  level,  is  upon  a bench  or  turning-point,  and  is  a “ + 
sight  ; ” all  other  readings  are  “ — sights/' 

III.  The  + sight,  added  to  the  height  of  the  point  upon 
which  the  reading  was  taken,  gives  the  “ Height  of  Instru- 
ment.” 

IV.  The  — sights  taken,  at  any  position  of  the  level, 
subtracted  from  the  “Height  of  Instrument”  for  that 
position,  give  the  corresponding  “ Surface  Heights.” 

V.  All  the  -f  sight  readings,  and  the  last  — sight  of 
each  set , being  upon  benches  or  turning-points,  are  taken  to 
thousandths  of  a foot.  The  remaining  “ — sights  ” are 
taken  to  tenths  only. 

Note. — It  will  be  observed  that  when  the  column  of  “ surface 
heights”  is  complete,  the  second,  third,  and  fourth  columns  of 
the  field-notes  are  no  longer  needed.  The  first  and  fifth  columns, 
which  together  contain  the  horizontal  and  vertical  measurements 
for  the  line  of  work,  afford  all  the  data  necessary  for  mapping 
the  profile  and  determining  the  grade-line. 

306.  The  location  of  the  benches  should  be  so  described  in 
the  column  of  “remarks,”  that  any  particular  bench  may  be 
found  at  any  time,  by  referring  to  the  field-notes.  The  impor- 
tance of  this  is  apparent  when  it  is  remembered  that  the  process 
of  construction  destroys  or  removes  the  stakes  along  the  line  of 
the  survey,  and  that  the  question  of  the  completion  of  the  work 
can  be  determined  only  by  reference  to  the  benches.  It  is 
obvious,  also,  that  they  should  be  established  somewhat  off  the 
line  of  the  survey.  The  distance  apart,  of  regularly  established 
benches,  should  be  governed  by  the  above-mentioned  uses  of 
them. 

Any  turning-point  may  be  profitably  made  a bench  (when 
it  can  be  made  permanent),  by  carefully  recording,  so  as  to 
admit  of  its  identification. 


SEC.  IV.] 


SECTION  LEVELING. 


269 


In  order  to  eliminate  the  effects  of  curvature  and  refraction, 
‘‘turning-points,”  whether  used  as  benches  or  not,  should  be 
taken  at  equal  distances  from  the  instrument  when  practicable, 
or  compensation  should  be  secured  by  proportional  distances,  as 
in  Art.  297. 

307.  In  conducting  a section  level  through  a rocky  district, 
turning-points  in  abundance  are  found  at  hand,  and  cause  no 
delay  in  their  preparation,  whereas  a bench  in  the  same  section, 
requires  marking  and  locating. 

In  leveling  through  flat  and  level  sections  of  country, 
although  the  engineer  can  get  “sights”  for  long  distances,  a 
proper  regard  for  accuracy  will  induce  him  to  limit  the  distance, 
between  successive  positions  of  the  level,  to  about  six  hundred  feet. 
Under  such  circumstances,  each  turning-point  is  made  a bench. 

308.  The  methods  of  establishing  benches  are  various.  In  a 
rocky  section,  some  conspicuous  point  is  marked  either  by 
drilling  or  grooving  the  rock.  In  villages  or  cities,  stone  steps, 
or  projecting  courses  of  masonry  to  dwellings,  curb-stones,  and 

.fence-posts  afford  good  benches,  and  admit  of  easy  identification. 

In  sections  where  trees  abound,  a notch  is  cut  in  the  side 
of  a trunk  near  the  root,  in  such  a manner  as  to  leave  a pro- 
jecting point  upon  which  the  rod  may  be  held  vertically.  A nail 
driven  full  length  into  the  projection,  gives  it  the  necessary  firm- 
ness for  a bench. 

In  marshes  or  prairies,  where  there  are  neither  rocks  nor 
trees,  the  engineer  is  compelled  to  resort  to  long  stakes,  firmly 
driven  into  the  ground  to  such  a depth  as  to  be  undisturbed  by 
the  frost ; no  portion  of  the  stakes  being  allowed  to  project  above 
the  surface.  The  top  of  each  is  trimmed  to  a kind  of  blunt 
point,  into  which  a nail  is  driven  its  full  length. 

A re-survey  of  a route,  to  detect  possible  errors  in  leveling, 
is  accomplished  by  taking  the  heights  of  the  “benches”  only, 
and  is  called  a “check  level.” 


270  ELEMENTS  OF  SURVEYING.  [ HOOK  VII. 

Drawing  the  Profile. 

309.  When  the  “section  level  ” of  a 1 i no  of  work  lias  been 
completed,  the  “ profile”  is  next  to  be  drawn.  The  method  of 
doing  this  is  very  simple. 

A horizontal  line  to  represent  the  datum  line  is  first  drawn, 
and  the  distances  from  the  first  column  of  notes  are  laid  off 
along  it,  to  a convenient  scale;  this  for  ordinary  working  draw- 
ings is  about  two  hundred  feet  to  an  inch. 

The  “ surface  heights”  corresponding  to  these  distances  are 
next  laid  off  at  right  angles  to  the  datum  line,  and  above  it, 
but  to  a scale  usually  ten  times  as  great  as  that  employed  for 
the  horizontal  distances;  that  is,  an  inch  upon  the  vertical  lines 
represents  one-tenth  as  many  feet  as  upon  the  datum  line.  A 
line  joining  the  upper  extremity  of  the  verticals,  is  the  profile. 

By  thus  employing  two  different  scales,  the  irregularities 
of  the  surface  are  made  more  apparent  to  the  eye,  and  the  sub- 
sequent adjustment  of  the  “grade-line”  is  rendered  much  easier 
and  more  accurate. 

310.  Every  earthwork  of  importance  requires,  in  addition  to 
the  working  profiles,  a general  map,  in  which  the  plan  drawn  from 
the  transit  survey  is  represented  upon  the  same  sheet  as  the  profile. 

The  horizontal  distances  of  both  portions  of  the  map  being 
drawn  to  the  same  scale,  and  one  being  placed  directly  above  the 
other,  corresponding  points  in  plan  and  profile  are  readily  compared. 

In  published  maps  of  this  kind,  representing  extended  works, 
and  drawn  for  convenience  to  a very  small  scale,  the  vertical 
scale  of  the  profile  is  frequently  several  hundred  times  as  great 
as  the  horizontal. 

Establishment  of  the  Grade. 

311.  The  determination  of  the  height  which  the  finished 
road  or  canal  shall  have  above  the  datum  line  at  different  points, 
is  called  “Establishing  the  grade.” 


SEC.  IY.J 


SECTION  LEVELING. 


271 


The  position  and  inclination  of  grade-lines  are  influenced  by 
a variety  of  circumstances : 

1st.  The  character  of  the  work.  A street  admits  of  an  in- 
clination of  five,  or  even  eight  feet  in  a hundred,  and  requires 
about  one  foot  for  its  drainage,  while  a rise  of  two  feet  in  a 
hundred  upon  a railroad  is  exceedingly  rare.  A canal  is,  of 
course,  level,  the  change  of  height  being  effected  by  abrupt 
transitions  at  the  locks. 

2d.  The  economy  of  construction.  It  is  desirable  to  make 
the  earth  excavated,  form  the  required  embankments,  or,  in 
the  language  of  the  engineer,  “ to  make  the  cuttings  balance  the 
fillings.” 

It  is,  however,  sometimes  more  economical  to  throw  away, 
or  “ make  a spoil  bank  ” of  the  earth  of  an  excavation,  than 
to  transport  it  the  required  distance  for  the  embankment. 
Embankments,  for  similar  reasons,  are  often  constructed  of 
earth  obtained  outside  of  the  road  limits  (“  borrowing  pits  ”) ; 
or,  when  such  means  are  not  available,  are  often  made  of  timber 
framing  (trestle-work). 

3d.  The  natural  obstacles,  which  render  the  construction 
difficult  ; such  as  rocky  ledges,  marshes,  lakes,  streams,  and 
quicksands. 

In  any  case,  the  engineer  determines,  by  inspection  of 
the  maps,  at  what  points  the  grade-line  shall  intersect  the 
natural  surface.  Thus  the  inclination  of  the  grade,  and,  con- 
sequently, its  height  above  the  datum  line,  for  each  “distance,” 
are  easily  found. 

Another  column  of  notes  is  now  made,  recording  these 
“ Grade  Heights ; ” each  being  placed  against  the  corresponding 
surface  height. 

312.  The  following  example,  with  its  accompanying  diagram, 
illustrates  the  method  of  establishing  a grade  and  recording 
the  notes.  It  will  be  observed  that  the  profile,  with  its  “ dis- 


272  elements  oe  surveying.  [book  VII. 

tances”  and  “ surface  heights,”  are  the  same  as  in  the  preceding 
problem. 

We  will  suppose  it  is  required  'to  establish,  in  the  following 
profile,  a grade-line  whose  inclination  shall  not  exceed  3 in  100  ; 
the  grade  to  begin  at  station  0,  at  the  surface. 


Dist. 

+s. 

H.  of  Ins. 

-S. 

H.  of  Sur. 

H.  of  Gr. 

Cut. 

Fill. 

Rem. 

0 

29.5 

29.5 

1 

29.8 

26.8 

3.0 

^60 

30.7 

25.2 

5.5 

2 

00 

24.1 

4.1 

Oat  287 

3 

20.8 

21.4 

0.6 

T.P. 

4 

16.3 

18.7 

2.4 

5 

13.1 

16.0 

2.9 

530 

11.7 

14.7 

3.0 

6 

12.3 

13.3 

1.0 

7 

11.5 

Oat  65a 

It  is  an  easy  matter  to  represent  any  required  inclination 
of  grade  on  the  profile  map;  nothing  more  being  necessary 
than  to  lay  o(T  the  proper  distances  on  two  different  verticals. 


SEC.  IV.] 


SECTION  LEVELING. 


273 


and  draw  a line  through  the  points  of  measurement.  For  in- 
stance : a grade  of  3 in  100,  running  downward  from  station  0, 
would  intersect  the  vertical  at  6,  eighteen  feet  lower,  and  the 
vertical  at  7,  twenty-one  feet  lower. 

Moreover,  by  consulting  the  notes,  we  find  that  a grade- 
line from  0,  whose  height  is  29.5  feet,  ending  at  the  surface 
at  7,  whose  height  is  11.5  feet,  descends  18  feet  in  700,  or 
2,57  in  100. 

Either  of  these  lines  would  fulfil  the  required  conditions. 
The  first  would,  however,  require  in  its  construction  a large 
excess  of  excavation  over  the  embankment  (as  may  be  seen  by 
drawing  a faint  line  in  the  diagram).  The  second  would  give 
an  excess  of  embankment. 

It  is  best,  generally,  that  the  cutting  should  be  slightly  in 
excess,  as  nearly  all  kinds  of  earth  shrink  a little  in  the  process 
of  removal. 

The  cuttings  and  fillings  of  the  profile  may  be  balanced 
with  tolerable  accuracy,  by  stretching  a thread  across  the  pro- 
file so  as  to  intersect  at  the  0 point,  and  then  varying  the  in- 
clination, until  the  areas  cut  off  by  the  profile  line  on  opposite 
sides  of  the  thread  appear  equal.* 

The  column  of  Grade  Heights  must  now  be  filled.  It  is 
easily  and  rapidly  done.  The  height  of  Grade,  at  0,  is,  by  the 
conditions,  29.5.  At  station  1,  it  must  be  2.7  lower,  or  26.8 ; at 
1.60,  4.3  lower,  or  25.2;  and  at  2,  5.4  lower,  or  24.1,  &c. 

The  remaining  columns  of  “cut”  and  “fill”  contain  simply 
the  differences  between  corresponding  “surface”  and  “grade 


* The  advantage  of  a thread  over  a ruler  lies  in  the  fact,  that  while  using  the  thread, 
the  areas  on  both  sides  of  it  are  seen  at  once. 

In  the  present  example,  a line  from  0,  descending  2.7  to  100,  seems  to  accomplish  the 
desired  purpose.  The  line  being  drawn,  the  “cut”  and  “fill”  areas  are  measured,  to 
determine  if  they  are  properly  balanced. 

The  complete  computation  of  the  earthwork,  by  which  the  exact  position  of  the  grade- 
line is  determined,  is  explained  in  a following  section. 


274 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


heights.”  Where  the  surface  is  higher  than  the  grade-line,  the 
construction  requires  a “ cutting ; ” when  the  established  grade- 
line is  higher  than  the  surface,  an  embankment,  or  “ filling,”  is 
necessary. 

The  notes  in  the  final  column,  indicate  the  points  where  the 
grade-line  intersects  the  natural  surface.  Such  are  called  zero 
points. 

The  distances  are  of  importance  in  the  computation  of  the 
earthwork.  The  above  notes  literally  signify  that  either  cut  or 
fill  is  0,  at  2.87,  also  at  6.52. 

These  distances  are  obtained  with  sufficient  accuracy  for 
ordinary  purposes  by  a measurement  of  the  profile  map.  When 
the  cuttings  and  fillings  are  recorded  in  the  proper  columns, 
the  notes  belonging  to  the  section-level  are  complete.* 

Note. — It  will  be  observed  that  the  first  set  of  notes  on 
page  265,  did  not  contain  the  columns  for  cut  and  fill. 

The  practice  in  keeping  the  notes  differs  with  the  work  to 
be  performed. 

In  extensive  railway  surveys,  it  is  convenient  to  rule  the 
pages  of  the  note-book  as  in  the  first  example ; carrying  out  the 
field-notes  to  the  extent  of  the  surface  heights,  at  least ; then 
transfer  to  another  book,  the  “distances,”  “surface  heights,”  and 
“grade  heights,”  ruling  columns  for  “cut,”  “fill,”  and  “remarks.” 


* The  following  calculation  may  be  employed  in  the  more  important  cases.  The 
triangles  formed  by  the  verticals  (cut  or  fill),  the  grade-line,  and  the  surface-line  are 
similar,  and  give  the  following  proportion  : 

The  sum  of  the  cut  and  fill, 

: the  cut, 

: : the  distance  from  cut  to  fill, 

■ distance  from  the  cut  to  0 point. 

Fill  may  bo  substituted  for  cut  in  the  second  and  fourth  terms. 

The  application  to  the  first  zero  point,  in  the  above  notes,  is  as  follows : 

4.1 +0.0  : 4.1  ::  100  : required  diet.,  or  87. 

In  the  second  cuho  in  the  notes,  the  cut  necessary  to  the  calculation  is  wanting,  but 
la  easily  supplied,  by  determining  the  height  of  grade  in  the  usual  way  at  station  7. 


SEC.  IV.j 


SECTION  LEVELING. 


275 


These  transferred  notes  are  recorded  in  ink,  and  reserved  for 
use  in  mapping  and  computations. 

313.  Most  road  or  canal  surveys  are  made  on  several  trial- 
lines before  one  is  finally  adopted.  The  profile  of  each  line  is  care- 
fully drawn,  and  the  cost  of  construction  approximately  estimated. 

When  the  route  is  finally  selected,  and  the  section  levels 
satisfactorily  completed,  the  exact  width  of  the  earthwork,  both 
in  excavations  and  embankments,  is  carefully  staked  out  and 
the  amount  of  material  to  be  moved  in  the  progress  of  construc- 
tion, accurately  measured. 

The  method  of  conducting  this  work  is  explained  in  a follow- 
ing section. 


314.  Before  closing  the  subject  of  section  leveling,  we  will 
consider  the  profile  represented  in  the  figure,  and  the  set  of  field- 


notes  appended,  which  are  only  partially  completed,  and  which 
will  afford  some  examples  for  practice. 


276 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


Diet. 

+ s. 

Ht.  of  Ins. 

- S. 

Surface 

Heights. 

Item  arks. 

Bench. 

1.032 

0 

3.2 

1 

3.8 

2 

5.3 

3 

8.9 

35° 

10.3 

4 

9.0 

5 

4.8 

T.  P. 

11.815 

2.346 

T.  P. 

10.942 

2.318 

6 

9.7 

640 

6.4 

7 

2.1 

1.  What  is  the  “Height  of  Instrument”  for  the  first  position 

of  the  level  ? Ans.  31.032. 

2.  What  is  the  height  of  the  first  T.  P.  ? Ans.  28.686. 

3.  What  is  the  “Hfc.  of  Inst.”  for  the  second  position  of  the 

level  ? Ans.  40.501. 

4.  What  is  the  height  of  the  second  T.  P.  ? Ans.  38.183. 

5.  What  is  the  “Ht.  of  Inst.”  for  the  third  position  of  the 

level  ? Ans.  49.125. 


6. 

What  is  the  Height  of  Surface  at  0 ? 

Ans.  27.8. 

7. 

a a (( 

te 

at  3? 

“ 22.1. 

8. 

« t(  « 

(( 

at  350  ? 

“ 20.7. 

9. 

U ((  (C 

a 

at  5? 

“ 26.2. 

10. 

((  te  u 

. « 

at  6 ? 

“ 39.4. 

at  7? 

“ 47.0. 

11. 

Write  the  “ Surface 

Heights 

” for  the  distances  1, 

and  (J«. 


SEC.  V.j 


CROSS-SECTION  LEVELING. 


27? 


SECTION  V. 

CROSS-SECTION  LEVELING. 

315.  All  earthworks,  whether  excavation  or  embankment, 
unless  held  in  position  by  retaining  walls,  require  to  be  con- 
structed with  a sloping  surface,  the  inclination  of  which  depends 
upon  the  kind  of  earth. 

If,  in  a railway-cutting,  for  instance,  the  banks  which  bound 
it  be  left  too  nearly  vertical,  when  first  constructed,  the  weather- 
ing influences,  to  which  they  are  subjected,  soon  cause  the  ma- 
terial to  slide  down,  until  the  whole  slope  gradually  assumes  a 
much  lower  inclination. 

After  a time,  however,  the  tendency  to  roll  or  slide  is  checked 
by  the  friction  of  the  particles  themselves,  and  the  slope  thus 
formed  will  withstand  the  ordinary  effects  of  sun,  wind,  and 
rain.  The  inclination  thus  assumed  is  called  the  “ natural  slope” 
of  that  kind  of  earth.* 

316.  Slopes  are  expressed  mathematically  by  the  ratio  of 


S E E S' 


their  horizontal  to  their  vertical  dimensions,  and  which  is  called 
the  ratio  of  slope. 


* Thin  elope  is  determined,  experimentally,  by  drying  a portion  of  the  earth,  and 
then  pouring  it  from  a slight  elevation  upon  a level  surface.  The  heap  thus  formed 
is  a rather  flat  cone,  whose  sides  stand  at  the  lowest  inclination  they  would  be  liable  to 
assume  under  the  action  of  atmospheric  influences.  The  angle  with  the  horizontal  plane 
will  be  somewhere  between  26°  and  45°. 


278 


ELEMENTS  OF  SURVEYING. 


[HOOK  VII. 


In  the  diagram,  which  represents  a road-cutting,  the  ratio 
of  ES  to  AE,  or  of  FS'  to  BF ' is  the  ratio  of  slope. 

In  practice,  the  slope  at  which  earthworks  are  allowed  to 
stand  vary  from  1 to  1,  or  45°  (as  in  very  coarse  material),  to  2 
to  1,  or  26°  34',  in  very  fine  sand. 

A slope  of  to  1 (33°  41')  is  found  to  be  so  far  suitable 
for  all  ordinary  excavations  or  embankments,  that  it  is  common, 
in  the  absence  of  an  examination  of  the  material,  to  adopt  it 
as  the  ratio  of  slope  throughout. 

Setting  Slope  Stakes. 

317.  It  is  evident  that  the  width  of  natural  surface  of 
ground,  required  in  the  construction  of  a road,  will  vary  with 
the  depth  of  excavation  or  embankment. 

As  often,  therefore,  as  it  is  found  necessary  to  determine 
the  depth  of  the  cutting  or  filling,  in  the  section  level,  it  is 
also  necessary  to  mark  the  boundaries  of  the  width  of  the 
work,  on  the  natural  surface.  This  is  done  by  stakes  called 
Slope  Stakes , and  the  field-work  necessary  to  determine  their 
position,  and  to  measure  the  section  taken  across  the  road  of 
which  the  Slope  Stakes  indicate  the  boundaries,  is  called  “ Cross- 
Section  Leveling,”  or  “ Cross-Section  Work.” 

318.  A party  of  five  may  be  usefully  employed  in  setting 
Slope  Stakes ; viz.,  a leveler,  rodman,  axeman,  and  two  tape- 
men. 

The  rod,  for  cross-section  work,  is  a ruder  instrument  than 
that  employed  in  the  section  level.  It  should  be  at  least  fifteen 
feet  long,  with  the  feet  and  tenths  plainly  marked.  It  requires 
no  target,  the  leveler  himself  reading  the  rod  in  the  act  of 
sighting. 

The  field-book  is  ruled  as  shown  below. 


SEC.  V.] 


CROSS-SECTION  LEVELING. 


279 


Dist. 

Left. 

Centre 

cuttings. 

Right. 

The  left-hand  column  contains  the  distances  taken  from 
the  section-level  notes.  The  third  column  is  for  the  cut  or 
fill,  corresponding  to  the  distance  in  the  first  column ; these 
numbers  also  being  taken  from  the  notes  of  the  section  level. 

A filling,  it  should  be  remarked  here,  is  designated  as  a 
minus  cutting  in  the  field-notes. 

The  second  and  fourth  columns  are  for  the  horizontal  and 
vertical  measurements  of  the  cross-section. 

319.  The  examples  following  will  illustrate  the  method  of 
measuring  the  section  and  recording  the  notes  : 


Let  Figure  147  represent  a section  across  a road  excavation. 
AB  being  the  bed  of  the  road,  and  SOS'  the  line  of  the  natural 
surface. 

The  road-bed  is  supposed  to  be  16  feet  wide,  the  centre 
cutting  12.4  feet,  and  the  ratio  of  slope  1J  to  1. 

The  level  being  set  up  and  adjusted  in  a convenient  place,  the 
rod  is  first  held  by  the  centre  stake  at  C,  and  a reading  taken. 


280 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


In  the  present  example,  the  reading  is  5.4.  The  line  AB 
forms  a convenient  datum  line,  and  the  height  of  the  instru- 
ment above  this  line,  is  evidently  12.4  + 5.4  = 17.8  ft. 

This  is  noted  down,  for  the  moment,  on  a reversed  page 
of  the  note-book,  or  on  a spare  slip  of  paper  ; neither  the  height 
of  instrument  or  rod  readings  being  matters  of  permanent  record 
in  cross-section  work. 

It  is  evident  that  if  the  rod  be  held  at  different  points  along 
the  surface,  and  the  readings  subtracted  from  the  “height  of 
instrument,”  the  remainders  will  be  the  heights  of  these  points 
above  the  datum  line  AB.  These  heights  are  technically  called 
cuttings,  although  in  the  case  of  iS^and  S'V,  no  actual  excava- 
tion is  proposed. 

The  reading  at  E is  supposed  to  be  5.5.  The  cut  is  therefore 
17.8  — 5.5  = 12.3.  The  horizontal  distance  from  the  centre  is 
8 feet.  For  each  cutting  there  will  be  a horizontal  measurement, 
and  these  two  must  be  recorded  together. 

The  form  adopted  is  that  of  a fraction  in  which  the  numera- 
tor is  the. cutting  and  the  denominator  the  distance. 

The  record  of  this  measurement  would  be,  therefore, 
in  the  column  marked  “left.”  The  points  A and  B,  of  the 
cross-section  are  appropriately  termed  the  angles,  and  as  the 
points  E and  F,  directly  over  them,  become  new  starting-points 
for  horizontal  measurements,  it  is  important  to  distinguish  them, 
in  some  way,  in  the  notes. 

(Right  and  left  in  the  actual  survey  are  determined  by  the 
direction  in  which  the  survey  progresses,  and  in  which  the 
centre  stakes  arc  numbered.) 

A common  method  is  the  one  adopted  in  our  notes — to  sub- 
stitute for  the  number  which  represents  the  half  width  of  the 
road,  the  letter  A. 

The  hind-chainman  now  lakes  his  position  at  E,  and 


SEC.  V.] 


CROSS-SECTION  LEVELING. 


281 


the  remaining  distances  to  the  left  are  measured  from  this 
point. 

A change  in  the  surface-line  at  H requires  notice.  The 
reading  of  the  rod  6.2  indicates  a cut  of  11.6.  This,  with  the 
distance  from  E , 10  feet,  is  duly  recorded. 

There  being  no  other  material  change  in  the  surface  line 
beyond  H , there  remains  to  be  determined  on  this  side  only  the 
intersection  8,  of  the  surface  and  slope. 

It  is  found  by  trial.  When  found,  it  is  evident  that  the 
ratio  of  the  distance  to  the  “ cut,”  must  be  the  same  as  the  ratio 
of  slope.  In  the  present  example,  the  distance  must  be  1J  times 
the  cut. 

Suppose  a trial  reading  taken  at  25  feet  out,  is  3.2.  The 
height,  or  cut,  is  (17.8—3.2)  = 14.6.  1J  times  this  is  only  21.9 
feet.  The  distance  tried,  25  feet,  is  too  great. 

Suppose  a second  trial  at  22  feet  out,  with  a rod  reading 
of  3.8.  The  cut  is  14.  1^  times  this  is  21.  Still  too  far  out. 

A third  trial,  at  21  feet  out,  and  a reading  of  4,  gives  a cut 
of  13.8  ft.  This  multiplied  by  1J-  gives  20.7  feet.  The  measured 
distance  is  slightly  too  great,  but  in  ordinary  practice  this  ap- 
proximation would  be  considered  near  enough.  The  record  for 
the  slope-stake  8 would  therefore  be  ;f . * 

In  proceeding  from  the  centre  to  the  right,  we  find  a point 
K between  the  centre  and  the  angle,  that  requires  attention. 

The  distance  in  such  a case  is  taken  from  the  centre  instead 
of  the  angle.  CK  is  3 ft.  and  the  rod  reading  5.3  gives  a cut  of 
12.5.  The  reading  at  the  angle-stake  F is  5.8,  giving  a cut 
of  12  feet. 

If  the  surface-line  from  F were  level,  the  distance  FS'  would 
be  12  x 1^  = 18  feet ; but  as  the  ground  descends,  the  distance 
is  less. 

* The  readings  and  distances  in  this  example  have  been  made  to  correspond  to  a rise  of 
one  foot  in  five  from  II  to  S.  The  exact  record  for  S is  . 


282 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


A trial  at  11  ft.  with  a reading  of  the  rod  of  10.4,  indicates  a 
cut  of  7.4.  This  multiplied  by  1J  gives  11.1,  which  is  very  nearly 
right ; y7^  is  therefore  the  record  for  the  location  of  S'. 

The  completed  notes  for  this  cross-section  are  as  follows : 


Dist. 

Left. 

Centre  Cut. 

Right. 

13.8  11.6  13.3 
30i7  10  A 

12.4 

12.5  12  7.4 

3 A lTl 

320.  We  will  now  give  a similar  example,  illustrating  the 


method  of  staking  out  embankments : 

T TV  C H V 


Dist. 

Left. 

Centre  Cut. 

Right. 

— 15.8  —12.8 

— 11.7 

—8.6  —8.4 

A 12.6 

23.7  A 

The  filling  at  the  centre  is  assumed  to  be  11.7  ft.,  which 
appears  in  the  column  of  “centre  cut,”  with  its  appropriate 
sign. 

The  reading  of  the  rod,  at  the  centre,  as  shown  by  the  dia- 
gram, is  7.2. 


sec.  y.J 


CROSS-SECTION  LEVELING. 


283 


The  sum  of  the  reading  and  the  centre  cut  (7.2  — 11.7) 
is  —4.5,  which  is  the  “height  of  instrument”  referred  to 
the  line  AB. 

The  readings  at  all  other  points  along  the  line  SS'  must 
be  subtracted  from  this  “height  of  instrument,”  as  in  the  pre- 
ceding example.  The  several  remainders  are  the  corresponding 
“ cuttings.” 

The  reading  at  angle  stake  E , 8.3,  subtracted  from  — 4.5, 
gives  —12.8,  for  the  cut. 

For  the  slope  stake  S,  we  will  suppose  a trial  distance  of  20 
feet  from  E , and  a rod  reading  at  the  trial  point  of  10.8  feet. 
The  cut  is  therefore  —15.3;  this  multiplied  by  gives 
22.95.*  The  trial  distance  therefore,  20  feet,  is  not  enough. 

A trial  of  24  feet  out,  we  will  suppose  to  give  a rod  reading 
of  11.3  ft.,  which  corresponds  to  a cut  of  —15.8  ft.  The  ratio 
applied  to  this,  gives  for  the  proper  corresponding  distance  out, 
23.7  ft.,  which  is  nearly  correct.  The  distance  at  which  the  trial 
was  made  is  slightly  too  great.  It  is  evident  that  if  the  slope 
stake  be  set  at  the  calculated  distance,  23.7  ft.,  the  record  of 

— may  ^e  ma(^e  without  involving  an  error  of  more  than 
23.7 

a tenth  of  a foot,  in  either  cut  or  distance. 

On  the  right,  the  reading  at  the  angle  stake  F is  4.1.  The 
cut,  therefore,  is  — 8.G  feet. 

As  the  surface  rises  but  little  from  F to  S'  the  trial  distance 
for  the  slope  stake  is  taken  at  12  feet ; (it  should  be  12.9  ft.,  if 
the  surface  were  level) ; the  reading  is  supposed  to  be  3.9  feet. 
This  gives  the  cut  —8.4,  which  should  correspond  to  a distance 
out  of  12.6  feet.  It  is  evident,  that  considering  the  rise  in  the 
surface  and  the  rod  readings  at  F and  S' , the  rod  reading  at  S' 

* The  sign  is  disregarded  in  the  product.  It  may  he  well  to  notice,  however,  that  the 
ratio  of  slope  in  embankments  is  considered  to  be  lj  to  —1. 


284 


ELEMENTS  OF  SURVEYING. 


[ROOK  VII. 


would  not  vary  a tenth  if  moved  from  12  feet  to  12.G.  The 

—8.4 

record,  therefore,  for  S is  — — 

1/V.  O 


Note. — It  is  not  necessary  to  take  rod  readings  at  the  angle 
stakes.  If  desired,  these  readings  can  be  omitted  and  rod  read- 
ings taken  at  the  centre  stake,  and  right  and  left  from  it  at  each 
change  of  inclination  of  surface.  It  is  often  convenient,  how- 
ever, particularly  for  beginners,  to  take  the  angle  stake  readings, 
and  hence  in  the  rule  these  readings  are  assumed  to  be  taken. 


321.  The  rules  for  conducting  and  recording  cross-section 
work,  whether  for  excavation  or  embankment,  are  as  follows  : 

I.  Prepare  the  field-book  by  ruling  columns  for  Dis- 
tances and  Centre  Cuttings , leaving  wider  spaces  on  each 
side  of  the  latter  column  for  the  record  of  the  various 
measurements  to  the  left  and  right  of  the  centre  stake. 
Transfer  from  the  section-level  notes  the  distance  and 
corresponding  cut  or  fill,  for  each  stake  of  that  survey. 
Filling  in  the  cross-section  notes  is  designated  as  minus 
cutting. 

II.  Having  set  the  level  in  convenient  proximity  to  a 
proposed  cross-section,  take  a reading  of  the  rod  at  the 
centre-stake.  Add  this  reading  to  the  centime  cutting, 
( regarding  the  sign  of  the  latter),  to  obtain  the  “ height 
of  instrument .” 

1 1 T.  Lay  off  half  the  width  of  the  road-bed  each  side 
of  the  centre,  and  mark  the  distances,  temporarily,  with 
stakes.  These  are  the  angle  stakes. 


TV.  Proceed  to  take  rod \ readings  at  the  angle  stakes, 
and,  beyond,  them  outward  (on  a line  at  right  angles  to 
the  direction  of  the  line  of  the  road ),  at  each,  change  of 
inclination  of  the  surface.  Subtract  each  reading  from 


SEC.  V .] 


CROSS-SECTION  LEVELING. 


285 


the  height  of  instrument ; the  remainder  is  the  cutting, 
or  vertical  distance  of  the  point  measured,  from  the  pro- 
posed road-bed. 

V.  Record  each  cutting,  together  with  its  horizontal 
distance  from  the  nearest  angle  stake , in  the  form  of  a 
fraction  expressing  the  ratio  of  the  distance  to  the  cut- 
ting. Each  fraction  being  recorded  in  its  proper  column 
either  “ right  ” or  “ left  ” of  the  centre.  Points  between  the 
centre  and  angle  stake,  are  located  by  measurements  from 
the  centre. 

VI.  To  find  the  position  of  the  slope  stake : Measure 
off  a trial  distance  from  the  angle  stake,  and  determine 
the  cut  as  before.  Multiply  the  cut  by  the  ratio  of  height 
to  base  of  the  proposed  slope.  If  the  trial  distance  be 
greater  than  this  product,  the  assumed  point  is  too  far 
out,  and  vice  versa.  Repeat  the  trial  until  the  ratio  of 

the  distance  to  the  cut  expresses  the  ratio  of  slope. 

* 

322.  The  cutting  at  the  angle  stake  is,  in  cases  of  a tolerably 
uniform  surface,  a good  guide  to  the  distance  to  the  slope 
stake.  Thus,  when  the  angle  cutting  of  an  excavation  is  16 
feet  and  the  ratio  of  slope  1^  to  1,  the  distance  out,  for  a level 
surface,  would  be  24  feet ; but  if  the  ground  in  that  distance 
rise  2 feet  (and  which  in  practice  may  be  determined  pretty 
correctly  by  the  eye),  then  the  horizontal  distance  must  be  in- 
creased by  something  more  than  1J  times  2 feet.- 

When  the  surface  descends,  the  estimated  distance  out,  for 
a level  surface,  should  in  like  manner  be  diminished.  In  em- 
bankments the  conditions  are  reversed  ; the  steeper  the  rise,  the 
shorter  the  distance  out. 

323.  The  following  examples  will  serve  to  elucidate  the  sub- 
ject still  further: 


28G 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIE 


Diet. 

Left. 

Centre  Cut. 

Right. 

-4. 

-4.5 

—4.9 

— 6.1 

— 6.4 

-7.4 

(h 

A 

X 

4 

11.1 

Figure  148  represents  an  embankment  cross-section,  in 
which,  by  reason  of  the  small  depth  of  filling,  the  height  of 
instrument  is  a positive  quantity. 

The  centre  cut  is  —4.9  ; reading  of  the  rod  at  the  centre, 
8.5  ; the  sum  of  these,  or  “ height  of  instrument,”  is  3.6.  The 
remaining  rod  readings  are  given  on  the  line  through  the 
instrument. 

324.  In  the  example  of  the  following  diagram  (Fig.  149), 
the  cross-section  is  partly  in  excavation  and  partly  in  embank- 
ment. The  ratio  of  slope  is  2 to  1.  The  centre  cut  is  2.4. 
The  centre  reading  is  7.9  ; height  of  instrument,  10.3. 

The  reading  at  //,  is  6.3;  at  E,  6.1;  at  S,  5.4.  The 
point,  K,  is  easily  found  in  practice,  it  being  that  point  on 
the  surface  line  where  the  reading  of  the  rod  exactly  equals 
the  height  of  instrument.  The  reading  at  F is  13.2;  and 
at  S',  15.8. 

From  these  readings  the  cuttings  may  bo  found,  and  the 
notes  completed  as  below. 


SEC.  V.] 


CROSS-SECTION  LEVELING. 


287 


Dist. 

Left. 

Centre  Cut. 

Right. 

4.9 

4.2 

4.0 

2.4 

0 

-2.9 

-5.5 

9.8 

A 

3^8 

.4 

A 

11 

325.  In  the  following  example,  there  is  a regular  rise  in  the 
surface-line  of  one  foot  in  eight.  The  ratio  of  slope  in  the 
excavation  is  to  be  1 J to  1 ; height  of  instrument,  14.2. 

In  seeking  for  the  position  of  the  slope-stake  S’,  a distance 
out  of  13  feet  is  tried  ; the  reading  of  the  rod  at  the  trial 
point  is  4.8. 


S' 


Fig.  150. 

How  does  this  point  compare  with  the  true  position  of  $'? 

Ans.  Not  far  enough  out. 

What  is  the  result  of  a trial  at  1G  feet  out  and  a reading 
of  4.4  ? Am.  Too  far  out. 

9.G 

What  is  the  true  cut  and  distance  at  S'?  Ans.  rrr 

14.4 


288 


ELEMENTS  OF  SURVEYING. 


[book  VIL 


Find  the  position  of  8. 


Ans. 


Note  1. — It  sometimes  happens  in  very  hilly  sections,  that 
it  is  impracticable  to  sight  to  all  the  necessary  points  of  a 
single  cross-section  from  one  position  of  the  level.  In  such 
a case,  it  is  only  necessary  to  work  from  the  centre  as  far  as 
the  surface  will  permit,  then  establish  a turning-point,  precisely 
as  in  section  leveling;  change  the  position  of  the  level  so  as 
to  proceed  with  the  work,  and  determine  the  new  height  of 
instrument,  from  which  the  readings  are  to  be  subtracted  as 
before. 


Note  2. — The  degree  of  accuracy  desirable  to  be  attained 
in  setting  the  slope  stake,  varies  with  the  kind  of  earth  to  be 
“ staked  out,”  so  that  no  exact  rule  can  be  laid  down. 

A principle,  in  quite  general  nse,  permits  the  stake  to  be 
set  when  the  calculated  distance  varies  from  the  trial  distance 
by  less  than  a foot. 

The  limit  of  error  should  never  be  greater  than  this,  but 
in  rock  and  the  harder  kinds  of  earthwork,  it  should  be  made 
much  less. 


SECTION  VI. 

COMPUTATION  OF  EARTHWORK. 

326.  Before  the  work  of  construction  of  a railroad  or  canal 
commences,  the  calculation  of  the  earthwork  must  be  com- 
pleted. 

The  cross-section  levels  afford  the  necessary  data.  These 
surveys  have  divided  the  proposed  work  into  blocks  of  100  feet, 
or  less,  in  length,  and  which  are  appropriately  termed  prismoids. 


SEC.  VI.] 


computation  oe  earthwork. 


289 


Different  methods  are  employed  for  estimating  their  cubic  con- 
tents. The  most  accurate,  though  the  most  laborious,  is  the 
prismoidal  formula  (Leg.,  Mensuration,  page  129), 

vol.  = + + 

B and  B'  representing  the  areas  of  the  end  sections  of  the 
prismoid,  M the  area  of  a section  midway  between  them,  and 
l the  entire  length  of  the  solid. 

The  principal  difficulty  in  applying  this  formula  lies  in 
finding  the  dimensions  of  the  middle  section. 

327.  We  will  show  the  application  of  the  formula  by  an 
example  of  road  excavation. 

To  simplify  the  problem,  we  will  suppose  such  a degree 
of  regularity  in  the  ground  surface  that  the  angle  cuttings 
may  be  omitted. 

The  length  is  supposed  to  be  100  feet.  The  other  dimen- 
sions are  given  in  the  diagram. 

The  areas  of  the  end  sections  are  easily  found.  It  is  only 
necessary,  in  each  case,  to  add  together  the  areas  of  the  trape- 
zoids composing  the  whole  end  figure,  as  represented  in  the 
diagram,  and  subtract  therefrom  the  sum  of  the  triangles  which 
lie  outside  the  section.  The  dimensions  of  these  triangles  are 
always  expressed  in  the  cross-section  notes,  by  the  records  for 
the  slope  stakes. 

The  area  of  B is  thus  found  to  be  104.8  sq.  feet,  and  of  i?', 
116  sq.  feet. 

Now,  if  a section  of  this  prismoid  be  taken  midway  between 
the  two  ends,  each  of  its  several  dimensions  must  be  an  arith- 
metical mean  of  the  corresponding  measurements  of  the  end 
sections.  Thus,  the  centre  cutting  is  found  to  be  5 ft.;  the  dis- 


290 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


tance  from  the  angle  to  slope  stake,  on  the  left,  9.9  ft., 


the  cutting  on  the  extreme  left  is  6.6  ft 


Fig.  151. 


The  area  of  if  is  111.3  sq.  ft. 

Vol.  of  the  prismoid  = i x 100  (104.8  -f  116  -f-  445.2), 

= 11100  cubic  feet. 

328.  In  applying  the  prismoidal  formula  to  an  example  in 
which  one  end  section  has  more  given  dimensions  than  the  other, 
the  calculator  is  frequently  in  doubt  how  he  shall  average  these 
dimensions  to  obtain  the  middle  section.  As  a rule,  each  cut- 
ting of  the  most  irregular  section  should  be  averaged  with  the 
cutting  nearest  opposite  to  it  in  the  other  section. 

We  will  illustrate  this  by  an  example;  representing  the  sec- 
tions by  the  field-notes  only. 


SEC.  VI.] 


COMPUTATION  OF  EARTHWORK. 


29] 


i Diet. 

i 

Left. 

Centre  Cut. 

Right. 

2 

17.2 

16.8 

16 

15.8 

13 

10  8.4 

25.8 

A • 

2 

A 

8'  12.6 

2.60 

11.6 

11.2 

10.4 

10 

8 

17+ 

A 

A 

12 

The  half-width  of  the  road,  for  which  A is  given  in  the  notes, 
is  to  be  considered  as  8 feet.  The  length  of  the  prismoid  is 
expressed  by  the  difference  of  the  given  distances,  or  60  feet. 

The  dimensions  of  the  middle  section  are  found  as  follows : 
The  centre  cut  is  half  the  sum  of  the  given  similar  dimen- 


sions, 


15.8  + 10.4 
2 


= 13.1 


feet. 


On  the  right,  the  average  of  the  angle  cuttings  gives  ; 

for  the  next  measurement,  both  cut  and  distance  must  be  aver- 
aged ; it  is, 

i X (10  + 8)  = 9 9_ 

% X (8  + 12)  = 10  10 


The  last  term  in  the  upper  section  must  be  averaged  with 
the  last  in  the  lower,  thus : 


|x(  8 + 8.4)  = 8.2  jh2 
| x (12  + 12.6)  = 12.3  °r  12.3' 

X 6 

On  the  left,  the  measurement  — of  the  upper  section,  must 

/V 

be  averaged  with  the  centre  cutting  of  the  lower,  being  nearest 
opposite  to  that  point.  We  have, 

\ X (16  + 10.4)  = 13.2  13J2 

*x(  2+  0 )=  1 0r  1 ' 

14 

At  the  angle,  in  like  manner,  we  have  ^ ; and  finally  at 
14  4 

the  slope  stake  tr— — • The  complete  dimensions  being 

21. 6 


292 


ELEMENTS  OF  SURVEYING. 


[BOOK  VII. 


14.4  14  13.2 
2L6  A 


11.5  9^  &2_ 

~A~  TO  12^3 


The  area  of  the  upper  section,  after  subtracting  the  triangles, 
as  before,  is  543.47  sq.  feet. 

The  area  of  the  lower  end  section  is  325.44. 

The  middle  section  contains  429.8  sq.  feet. 

The  volume  of  the  prismoid  is 

i x 60  (543. 47  + 325. 44  + 4 x 429. 8) 

= 25881.1  cubic  feet,  or  958.56  cubic  yards. 


329.  The  following  method*  of  computing  the  contents  of 
consecutive  volumes  between  regular  cross-sections  of  excavation 
and  embankment,  contained  by  uniform  slopes,  will  be  found  of 
service. 

The  accompanying  diagram  represents  the  cross-section  of  a 
railroad  cut,  b being  half  the  width  of  road-bed,  c centre-height, 


TV 


r elevation  of  right  slope-stake  above  grade,  r'  its  horizontal 
distance  from  nearest  side  of  road-bed,  l elevation  of  left  slope- 
stake,  V its  horizontal  distance  from  nearest  side  of  road-bed,  and 


* “ Fonnuluj  for  Railroad  Earthwork,”  by  J.  Woodbridge  Davis,  C.  E.,  Ph.  D. 


SEC.  VI.] 


COMPUTATION  OF  EARTHWORK. 


293 


w the  entire  top-width  or  horizontal  distance  between  slope- 
stakes.  The  area  of  this  section  is  evidently 

\rb  -f  \lb  -f  \c  (r'  + b)  + \c  (/'  -f  b). 

v l 

Let  S denote  the  ratio  of  slope ; then  8 = — , = j,,  and 

V i 

r — Sr',  l = SI'.  Substituting  and  reducing, 

Area  Section  = £ Sb  (r'  + l')  -}-  \c  (/  + /'  + 2b). 

Adding  and  subtracting  Sb2  does  not  change  its  value  ; 

Area  Section  — \Sb  (r'  + T + %b)  — Sb2-\-  \c  (r'-f  T + 25), 
or  Area  Section  = \w  (c-f Sb)  — Sb2. 

Supposing  w',  c'  to  represent  width  and  centre  at  next  station, 
the  area  of  its  cross-section  may  be  expressed  by  a formula  similar 
to  the  above;  half  the  sum  of  these,  multiplied  by  the  distance, 
D,  between,  and  divided  by  27,  gives  a near  approximate  of  the 
volume  in  cubic  yards  ; 

Vol.  = [wc  -f  w'c'  -f-  Sb  (w  + w') — 4 $£2] 

Add  two  consecutive  volumes  of  equal  length  by  means  of  the 
general  formula,  w",  c' , representing  the  width  and  centre  of 
third  cross-section : 

Vol.  = [wc  + 2 iv'c'  -f  w"c"  + Sb(w- f 2 w'  + tv")  - 8££2]  T^. 

By  continual  addition  we  may  get  a formula  for  the  sum  of 
any  number  of  consecutive  volumes  ; but,  letting  n denote  the 
number  of  volumes,  we  may  at  once  indite  a general  formula  for 
the  calculation  of  any  number  of  volumes  consecutive.  Thus 
we  have 

Vi— \ wc  + %iv'c'+  &c.  + 2wncn-f  wn+,cnfi  1 D 
° ‘ — ( + Sb(w  + 2w'  + &c.-\-2wD  + wn+x)  — £Sb2n  j 108 

Divide  and  multiply  by  2 to  convert  the  formula  into  more 
convenient  shape,  call  w'c',  w"c",  &c.,  mid-products,  wc  and 
Wn_,iCn+i  end-products,  and  we  have 


294 


ELEMENTS  OF  SURVEYING. 


[ROOK  VII. 


( mid-products  -f  | end-products 
Vol.  = < + Sb  (mid-widths  -f  | end-widths) 

( — 2Sb2  x no.  of  vols. 

Let  us  illustrate  this  formula  by  applying  it  to  the  following 
extract  from  a field-book,  containing  columns  of  stations,  centre 
cuts,  left  and  right  heights  and  distances  of  slope-stakes,  the 
road-bed  being  18  feet,  slope  1 to  1,  distance  apart  of  stations, 
100  feet : 


Station. 

Left. 

Centre. 

Right. 

1 

3.0 

T3°0 

2 

5.1 

ASr 

3 

6.4 

A-S 

4 

AS- 

7.2 

80 

TIO 

5 

m 

9.0 

A-^ 

6 

T633 

6.7 

i 

OPERATION. 


Stations. 

Widths. 

Centre. 

Products. 

1 

11.55 

X 

3.0  = 

34.65 

2 

27.8 

X 

5.1  = 

141.78 

3 

31.5 

X 

6.4  = 

201.60 

4 

34.1 

X 

7.2  = 

245.52 

5 

38.0 

X 

9.0  = 

342.00 

6 

15.85 

X 

6.7  = 

106.195 

9 or  Sbx  158.80  1071.745 

1429.2 

—810. 


9 ) 169094.5 
C ) 18788.3 


3131.38  cu.  yds. 


SEC.  VI.] 


COMPUTATION  OF  EARTHWORK. 


295 


330.  The  foregoing  method  of  calculating  earthwork  is 
approximate.  To  find  the  true  contents  we  use  a formula  of 
correction,  which  is  obtained  in  the  following  manner.  Let 

\w  (c  + Sb)  - Sb\  \w'  (c  + 86)  - S& 

be  the  end  areas  of  a volume  of  earthwork.  Then  \ (w  -f  id), 
y(c+ d)  are  evidently  the  width  and  centre  of  mid-section,  and 
its  area  is 

i(w+w')  +Sb)-SP. 


Multiply  this  by  4,  add  thereto  the  end  areas,  multiply  all  by  £ 
D,  and  divide  by  27,  to  obtain  in  cu.  yds. 


True  vol.  — 


'2  (wc  + w’c')  + ( xed  + id c) \ D 

. +aSb(w  + id')—128P  / 324 


This  formula  would  prove  very  unwieldly  to  carry  through  the 
calculations  for  series  of  volumes,  especially  in  the  consideration 
of  intermediate  stations.  The  same  results  may  be  obtained  in  a 
simpler  manner  by  using  the  difference  between  true  and 
approximate  contents  as  a correction.  Subtracting  the  approx- 
imate volume, 

\wc  + w'd  + S6(w- f w')—4:S62]  yj*, 
from  the  true,  we  have,  after  reduction, 

(w— w')  (d—c)D 
324  ’ 


for  the  error  or  correction. 


296 


ELEMENTS  OF  SURVEYING. 


HOOK  VII. 


Pris.  Cor. 

— 987 

— 481 

— 208 

- 702 

— 1449 


We  see  by  the  formula  that  to  correct  a volume, 
the  difference  of  widths,  found  by  a subtraction  in 
one  direction,  must  be  multiplied  by  the  difference 
of  centres  resulting  from  a subtraction  in  the 
opposite  direction,  this  product  multiplied  by  the 
length  of  volume  and  divided  by  324.  Applying 
12 ) 3827  this  rule  to  the  second  volume  of  example,  we 

9 ) 319  have  width  at  station  2 [27.8] — width  at  station  3 

3)35.4  [31.5]  = —3.7;  centre  station  3 [6.4]—  centre  sta- 

-11.81  Cor.  tion  2 [5.1]  = 1.3.  -3.7  x +1.3 

3131.38  Approx,  con’ts.  * 100  = -481.  Set  this  in  an 


3119.57  True  con'ts. 


extra  column.  Treat  each  volume 
in  like  manner,  remembering  that 
the  first  and  last  numbers  in  column  of  widths  are  half-widths. 
We  here  represent  the  column  of  corrections.  The  sum,  — 3827, 
divided  by  324,  is  —11.81  cu.  yds.  This  added  to  the  approx- 
imate result  yields  for  the  true  answer  3119.57  cu.  yds. 


It  may  happen,  in  practice,  that  some  of  the  volumes  involved 
need  no  correction  whatever,  which  fact  will  be  apparent  from 
the  formula  by  inspection  and  without  actual  labor. 


BOOK  VIII. 

TOPOGRAPHICAL  SURVEYING. 


331.  Besides  the  surveys  that  are  made  to  determine  the 
area  of  land  and  the  relative  positions  of  objects,  it  is  frequently 
necessary  to  make  minute  and  careful  examinations  for  the 
purpose  of  ascertaining  the  form  and  accidents  of  the  ground, 
and  to  make  such  a plan  as  will  distinguish  the  swelling  hill 
from  the  sunken  valley,  and  the  course  of  the  rivulet  from 
the  unbroken  plain. 

This  branch  of  surveying  is  called  Topography.  In  surveys 
made  with  a view  to  the  location  of  extensive  works,  the  de- 
termination of  the  slopes  and  irregularities  of  the  ground  is 
of  the  first  importance;  indeed,  the  examinations  would  other- 
wise be  useless. 

332.  The  manner  of  ascertaining  these  irregularities  is,  to 
suppose  the  surface  of  the  ground  to  be  intersected  by  a 
system  of  horizontal  planes  at  equal  distances  from  each  other ; 
the  curves  determined  by  these  secant  planes,  being  lines  of 
the  surface,  will  indicate  its  form  at  the  places  of  section, 
and,  as  the  planes  are  nearer  or  more  distant  from  each  other, 
the  form  of  the  surface  will  be,  more  or  less,  accurately  ascer- 
tained. 

If  such  a system  of  curves  be  determined,  and  then  pro- 
jected or  let  fall  on  a horizontal  plane,  it  is  obvious  that  the 
curves  on  such  plane  will  be  nearer  together  or  farther  apart, 
as  the  ascent  of  the  hill  is  steep  or  gentle. 


298 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIII. 


If,  therefore,  such  intersections  are  made,  and  the  curves  so 
determined  are  accurately  delineated  on  paper,  the  map  will 
give  such  a representation  of  the  ground  as  will  show  its  form, 
its  inequalities,  and  its  striking  characteristics. 

333.  The  subject  divides  itself,  naturally,  into  two  parts : 

1st.  To  make  the  necessary  examinations  and  measurements 
on  the  field  ; and, 

°d.  To  make  the  plot  or  the  delineations  on  paper. 

When  the  area  is  extended  and  the  contour  planes  are  widely 
separated,  points  along  the  ridges  and  summits,  and  also  along 
the  valley  bottoms,  are  located  by  the  Transit  or  Plane  Table, 
and  their  heights  above  a datum  plane  are  determined  by  means 
of  Transit  angles  of  inclination,  or  by  the  Y Level,  or,  in  rapid 
reconnaissance,  by  the  Barometer. 

We  shall,  perhaps,  be  best  understood,  by  giving  an  example 
or  two,  and  then  adding  such  general  remarks  as  will  extend 
the  particular  cases  to  others  that  may  occur. 

EXAMPLE  FIRST. 

334.  Let  A,  Fig.  153,  be  the  summit  of  a hill,  the  contour 
■of  which  it  is  required  to  determine  and  represent.  At  A,  let  a 
stake  be  driven,  and  let  the  axis  of  the  transit,  or  level,  be 
placed  directly  over  the  nail  which  marks  its  centre.  From  A, 
measure  any  line  down  the  hill,  as  AB,  using  the  telescope  of 
the  transit,  or  level,  to  arrange  all  its  points  in  the  same 
vertical  plane.  Great  care  must  be  taken  to  keep  the  measuring 
chain  horizontal,  for  it  is  the  horizontal  distances  that  are  re- 
quired. At  different  points  of  this  line,  as  a,  b,  c , d,  &c.,  let 
stakes  he  driven,  and  let  the  horizontal  distances  Aa,  ab,  be, 
and  cd,  bo  carefully  measured.  In  placing  the  stakes,  reference 
must  he  had  to  the  abruptness  of  the  declivity,  and  the  accuracy 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


299 


with  which  the  surface  is  to  be  delineated ; their  differences  of 
level  ought  not  to  exceed  once  and  a half,  or  twice,  the  distance 
between  the  horizontal  planes  of  section. 

Having  placed  stakes,  and  measured  all  the  distances  along 
the  line  AB , run  another  line  down  the  hill,  as  AC,  placing 
stakes  at  the  points  e,  /,  g , and  li,  and  measuring  the  horizontal 
distances  Ae,  ef,  fg,  and  gh.  Run  also  the  line  AD,  placing 
stakes  at  i,  l,  m,  and  n,  and  measuring  the  horizontal  distances 
Ai,  il,  Im,  and  mn. 

Each  line,  AB,  AC,  AD,  running  down  the  hill,  from  A, 
may  be  regarded  as  the  intersection  of  the  hill,  by  a vertical 
plane ; and  these  secant  planes  are  to  be  continued  over  all  the 
ground  which  is  to  be  surveyed.  If  the  work  is  done  with  a 
transit,  or  with  a level  having  a compass,  the  angles  DAB 
and  BAC,  contained  by  the  vertical  secant  planes,  can  be 
measured  ; if  it  is  done  with  a level,  having  no  needle,  let  any 


300 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIII. 


of  the  distances  ae,  bf,  ai,  bl , &c.,  be  measured  with  the  chain, 
and  there  will  then  be  known  the  three  sides  of  the  triangles 
Aae,  Abf ‘ Aai,  Abl,  &c. 

Let,  now,  the  difference  of  level  of  all  the  points  marked  in 
each  of  the  lines  AB , AD , AC,  be  determined,  being  careful  to 
hold  the  rod  upon  a point  near  each  stake  which  represents 
the  general  surface,  and  not  at  the  bottom  of  a hole  nor 
upon  the  top  of  a mound,  a precaution  to  be  observed  in  every 
leveling  operation. 

Let  now  the  heights  of  all  the  points  marked  on  each  of  the 
lines  AB,  AD,  and  AC,  be  found  with  reference  to  the  datum 
plane,  which,  near  the  coast,  may  be  the  plane  of  mean  low 
water,  or  a plane  assumed  so  that  it  will  lie  below  the  lowest 
point  to  be  delineated. 

In  the  present  example,  of  a single  slope  only,  the  results  of 
the  measurements  and  leveling  are: 


Line  AB. 


Distances. 

A a — 40  feet. 
ab  = 50  “ 

be  — 30  “ 
cd  = 46  “ 


Heights  above  datum  plane. 


A = 64  feet. 
a = 52  “ 

b = 44  “ 
c — 35  “ 

d — 24  “ 


Line  AC. 


Distances. 
Ae  = 28  feet. 
ef  = 45  “ 
fg  = 55  “ 

(jh  = 49  “ 


Heights  above  datum  plane. 


A = G4  feet. 
e = 53  “ 
/ = 44  “ 
g = 32  “ 
h = 18  “ 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


301 


Line  AD. 


Distances. 

Ai  = 25  feet. 
il  =55  “ 

lm  = 38  “ 

mn  = 48  “ 


Heights  above  datum  plane. 


A = 64  feet. 
i = 55  “ 

l = 42  “ 

m = 35  “ 


w = 21  “ 

Angle  CAB  = 25°,  Angle  DAB  = 30°. 

These  data  are  sufficient,  not  only  to  find  the  intersections 
of  horizontal  planes  with  the  surface  of  the  hill,  but  also  for 
delineating  such  curves  of  section  on  paper. 


Plot  of  Work. 


335.  Having  drawn,  on  the  paper,  the  line  A B,  lay  off  the 
angle  BAC  = 25°,  and  the  angle  BAD  — 30°.  Then,  from  a 
convenient  scale  of  equal  parts,  lay  off  the  distances  Aa , ab,  be, 
cd,  Ae,  ef,  fg , gh,  Ai,  il,  lm,  and  mn. 

Let  the  horizontal  planes  be  passed  at  distances  of  8 feet 
apart,  in  which  case  the  point  A,  in  the  example  given,  will 
lie  in  the  eighth  contour  plane,  counting  from  the  datum 
plane.  Since  A is  the  highest  point  of  the  hill,  and  the 
difference  of  level  of  the  points  A and  a,  is  12  feet,  the  first 
plane,  reckoned  downwards,  will  intersect  the  line  traced  on 
the  ground  from  A to  B,  between  A and  a.  Regarding  the 
descent  as  uniform,  which  we  may  do  for  small  distances,  without 
sensible  error,  we  have  this  proportion : as  the  difference  of  level 
of  the  points  A and  a,  is  to  the  horizontal  distance  Aa,  so  is  8 
feet,  to  the  horizontal  distance  from  A to  where  the  first  hori- 
zontal plane  will  cut  the  line  from  A to  B.  This  distance  being 
thus  found,  and  laid  off  from  A to  o,  gives  o,  a point  of  the 
curve  in  which  the  seventh  plane  intersects  the  ground.  The 
points  at  which  it  cuts  the  lino  from  A to  C,  and  the  line  from 


302 


ELEMENTS  OF  SURVEYING. 


[HOOK  VIII. 


A to  D , are  determined  similarly,  and  three  points  in  the  seventh 
curve  are  thus  found. 

The  graphic  operations  are  greatly  facilitated  by  the  aid  of 
a sectoral  scale  of  equal  parts,  of  which  Fig.  154  is  a repre- 
sentation. 


It  consists  of  two  arms,  or  sides,  which  open  by  turning 
round  a joint  at  their  common  extremity. 

On  each  arm  of  the  sector,  there  is  a diagonal  line  that 
passes  through  the  point  about  which  the  arms  turn : these 
diagonal  lines  are  divided  into  equal  parts. 

The  advantage  of  the  sectoral  scale  of  equal  parts,  is 
this : 

Let  it  he  proposed  to  draw  a line  upon  paper,  on  such  a 
scale  that  any  number  of  parts  of  the  line,  40  for  example, 
shall  be  represented  by  one  inch  on  the  paper,  or  by  any  part 
of  an  inch.  Take  the  inch,  or  part  of  the  inch,  from  the  scale 
of  inches,  on  the  sector;  then,  placing  one  foot  of  the  dividers 
at  40,  on  one  arm  of  the  sector,  open  the  sector  until  the 
other  foot  reaches  to  the  corresponding  number  on  the  other 
arm  ; then,  lay  the  sector  on  the  table  without  varying  the 
angle. 

Now,  if  we  regard  the  lines  on  the  sector  as  the  two  sides  of 
a triangle,  of  which  the  line  40,  measured  across,  is  the  base,  it 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


303 


is  plain,  that  if  any  other  line  be  likewise  measured  across  the 
angle  of  the  sector,  the  bases  of  the  triangles,  so  formed,  will  be 
proportional  to  their  sides.  Therefore,  if  we  extend  the  dividers 
from  50  to  50,  this  distance  will  represent  a line  of  50,  to  the 
given  scale ; and  similarly  for  other  lines. 

In  the  example  before  ns  (Fig.  153),  let  it  be  borne  in  mind, 
that  the  descent  from  A to  a , is  12  feet,  and  that  it  is  required, 
upon  the  supposition  of  the  descent  being  uniform,  to  find  that 
part  of  the  distance  corresponding  to  a descent  of  8 feet.  Take 
the  distance  from  A to  a , in  the  dividers,  and  open  the  arms  of 
the  sector  until  the  dividers  will  reach  from  12  on  the  line  of  equal 
parts,  on  one  side,  to  12  on  the  line  of  equal  parts  on  the  other. 
Then,  without  changing  the  angle,  extend  the  dividers  from  8 on 
one  side  to  8 on  the  other ; this  will  give  the  proportional 
distance  to  be  laid  off  from  A to  o.  Or,  if  the  dividers  be 
extended  from  4 to  4,  the  proportional  distance  may  be  laid  off 
from  a to  o. 

If  the  distances  to  be  taken  from  the  sector  fall  too  near 
the  joint,  let  multiples  of  them  be  used. 

336.  The  sixth  plane  is  to  pass  8 feet  below  the  first, 
that  is,  16  feet  below  A , or  4 feet  below  a,  a being  12  feet 
below  A.  Take  the  distance  ab,  in  the  dividers,  and  extend 
the  sector,  so  that  the  dividers  will  reach  from  8 to  (the 
descent  from  a to  b being  8 feet)  8,  or  from  80  to  80;  then, 
the  distance  from  4 to  4,  or  from  40  to  40,  being  laid  off 
from  a to  p , gives  p,  a point  of  the  sixth  curve. 

The  difference  of  level  between  a and  b being  8 feet,  and 
the  difference  of  level  between  a and  p being  4 feet,  the  dif- 
ference of  level  between  p and  b must  also  be  4 feet ; hence, 
the  fifth  plane  will  pass  4 feet  below  b,  and  q,  determined 
as  above,  is  a point  of  the  fifth  curve,  and  so  on.  After 
having  determined  the  points  in  which  each  contour  line  cuts 


304  elements  of  surveying.  [rook  viil 

the  lines  diverging  from  A,  let  the  contour  lines  be  drawn 
through  them,  so  as  to  indicate  the  surface  of  the  hill.  The 
numbers  (64),  (56),  &c.,  show  the  vertical  distances  of  the 
respective  planes  above  the  plane  of  reference. 

337.  Having  drawn  the  horizontal  curves,  the  next  thing  to 
be  done  is  so  to  shade  the  drawing  that  it  may  represent  accu- 
rately the  surface  of  the  ground.  This  is  done  by  drawing 
a system  of  small  broken  lines,  as  in  the  figure,  perpendicular 
in  direction  to  the  horizontal  curves  already  described.  In 
all  topographical  representations  of  undulating  ground,  the 
lines  of  shading  are  drawn  perpendicular  to  the  horizontal 
curves,  to  indicate  the  direction  of  the  flow  of  water  down  the 
declivity. 

A profile  along  either  of  the  diverging  lines  may  be  plotted 
by  the  rules  already  given.  Fig.  155  shows  the  profile  along 
the  line  AB. 


EXAMPLE  SECOND. 

338.  The  following  example  will  illustrate  the  methods  em- 
ployed in  making  a topographical  survey,  where  great  accuracy 
is  required. 

By  means  of  a transit  or  level,  range  a line  of  stakes  A,  B,  C, 
I),  E,  &c.,  Fig.  156,  along  one  side  or  through  the  middle  of  the 
ground  to  be  surveyed,  at  equal  and  convenient  distances  from 
each  other,  say  100  feet  apart.  Mark,  with  a piece  of  red  chalk, 
on  each  slake  in  this  row,  one  of  the  letters  of  the  alphabet, 


BOOK  VIII.]  TOPOGRAPHICAL  SURVEYING. 


305 


A,  B , C,  D,  E,  &c.,  in  their  order.  At  A,  range  a line  of  stakes, 
perpendicular  to  AE , planting  the  stakes  at  intervals  of  100 
feet;  and  mark  them  with  the  letters  A0,  At,  A2.  &c.,  which  are 
read  A zero,  A one,  A two,  &c. 

A 


33 


C 


D 


E 

At  B , range  a line  of  stakes  also  perpendicular  to  AE , and 
at  distances  of  100  feet  from  each  other,  and  designate  them 
1?0,  Bx,  B2,  &c.  Do  the  same  at  C,  D,  E , &c.,  until  all  the  stakes 
are  placed,  dividing  the  area  to  be  surveyed,  into  squares  of  100 
feet  on  a side.  The  letters  and  figures  should  be  plainly  marked 
on  a smooth  face  of  each  stake,  for  facility  of  reference.  If  this 
system  of  notation  be  followed,  the  stakes  may  be  recorded 
without  danger  of  confusion. 

339.  The  following  is  the  form  of  a field-book,  used  in  topo- 
graphical leveling: 


Fig.  156. 


306 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIII. 


Field  Notes. 


Benches. 

+ Sights. 

Height  Inst. 

— Sights. 

Stations. 

Heights. 

Bench 

4.136 

12.142 

12.1 

e2 

0.0 

1.9 

d2 

10.2 

C3 

11.906 

22.441 

1.607 

C3 

10.535 

6.0 

E, 

16.4 

6.9 

E3 

15.5 

6.8 

b3 

15.6 

1.9 

20.5 

3.7 

C2 

18.7 

6.8 

b4 

15.6 

1.7 

e4 

20.7 

b3 

11.914 

33.448 

0.907 

b3 

21.534 

8.0 

E. 

25.4 

4.0 

b2 

29.4 

4.1 

D„ 

29.3 

1.9 

C, 

31.5 

5.0 

As 

28.4 

9.9 

a4 

23.5 

1.7 

d4 

31.7 

1.1 

e4 

32.3 

0.1 

-^2 

33.3 

C„ 

11.813 

45.225 

0.036 

c„ 

33.412 

4.8 

B, 

40.4 

2.6 

B„ 

42.6 

A, 

8.925 

52.669 

1.481 

A, 

43.744 

3.2 

A„ 

49.5 

340.  Tn  the  example  taken  for  illustration  the  point  E2  is  the 
lowest  point. 

Set  up  the  level  and  take  a reading  upon  a bench,  which  has 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


307 


been  determined  to  be  8.006  feet  above  some  plane  of  reference, 
the  sea  level  for  instance.  Suppose  the  reading  to  be  4.136, 
which  is  entered  in  the  “ + Sight”  column.  The  “ Height  of 
Instrument”  will  therefore  be  12.142,  as  entered  in  the  proper 
column.  Next  take  readings  upon  E2,  D2,  and  C3,  reading  C3 
to  thousandths  as  it  will  be  used  as  a “ turning  point  ” (Art.  304). 
Remove  the  level  to  a new  position  and  read  C3  again,  11.906, 
which  gives  a new  “Height  of  Instrument”  22.441.  From  this 
position  read  as  many  stations  as  possible,  and  then  use  B3  as  a 
“ turning  point,”  and  so  on. 

Plotting  the  Work. 

341.  Draw,  on  a piece  of  paper,  a straight  line  AE.  From 
a scale  of  equal  parts,  set  off  distances  AB,  BC,  &c.,  each  to 
represent  100  feet.  Erect  perpendiculars  to  AE , at  each  of  the 
points  A,  B , C,  &c.,  and  then  set  off  the  distances  from  A to  1, 
from  1 to  2,  &c.,  each  to  represent  100  feet ; and  through  the 
points  1,  2,  3,  and  4,  draw  parallels  to  AE.  These,  by  their 
intersections  with  the  lines  drawn  through  A,  B,  G,  &c.,  will 
determine  the  position  of  the  stakes  A0,  A2,  &c.;  and  write  in 
red  ink  on  the  plot,  the  height  above  the  plane  of  reference  of 
each  stake,  taken  from  the  column  of  total  differences  in  the 
field-book.  Let  us  suppose  that  the  horizontal  planes  are  to  be 
taken  at  distances  of  6 feet.  We  may  find  the  points  in  which 
the  contour  lines  intersect  the  sides  of  the  rectangles,  as  in 
Example  First.  A very  convenient  scale  for  finding  the  points 
in  which  the  contour  lines  cut  the  sides  of  the  rectangles 
may  be  constructed  thus:  Upon  any  line  as  AB,  Fig.  157, 
erect  equidistant  perpendiculars  as  at  0,  1,  2,  &c.  Parallel 
to  AB  draw  lines,  alternately  heavy  and  light,  as  at  1,  2,  3,  &c. 
Suppose  we  wish  to  find  where  the  12-foot  plane  cuts  the  side  of 
a rectangle,  C3C\  for  example.  The  height  of  C3  is  10.5,  and 


308 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIII. 


of  (74  it  is  20.7,  from  which  data  we  find  the  rise  from  C3  to  CA 
to  be  10.2.  The  rise  from  03  to  the  12-foot  plane  is  1.5. 


13 

U 

n 

iio 0 

9 
8 

r 

6 
5 
4 
3 

3 

1 

B 

012  345678910 

Fig.  157. 

Now  fasten  a fine  thread  at  A on  the  scale,  and  stretch  it  to 
cut  10.2  on  the  vertical  scale,  as  AO  in  Fig.  157.  Look  along 
the  horizontal  line  1.5,  as  dotted,  and  its  intersection  with  AC 
will  be  at  S,  distant  16  ft.  on  the  horizontal  scale  from  A,  or  C3. 
The  tenths  on  the  vertical  and  horizontal  scales  may  be 
estimated  by  the  eye  with  sufficient  accuracy.  Engineer’s 
Section  Paper  may  be  advantageously  used  for  the  above 
purpose. 

342.  Tf  only  a rough  plot  is  needed,  the  Surveyor  may  take 
the  plot  thus  commenced,  into  the  field,  and  by  the  eye  trace 
(lie  contour  lines  on  the  map.  If  we  note  where  the  lines  of  the 
rectangles  cut  fences,  roads,  streams,  &c.,  we  can,  by  joining  the 
points,  obtain  a plot  of  the  ground. 


BOOK  VIII.]  TOPOGRAPHICAL  SURVEYING. 


309 


The  coutour  lines  may  be  traced  on  the  ground  as  follows : 
Set  up  the  instrument  and  read  a staff  placed  upon  a bench,  and 
determine  the  height  of  instrument  above  the  datum  plane.  Sup- 


pose it  to  be  11.432.  If  we  wish  to  mark  out  the  contour  line  six 
feet  above  the  datum  plane,  we  set  the  target  of  the  leveling  rod 
at  11.432 — 6 = 5.432,  and  direct  the  rodman,  by  signals,  up  or 
down  the  hill,  till  the  horizontal  wire  of  the  telescope  coincides 
with  the  horizontal  line  of  the  vane.  The  foot  of  the  staff  is 
then  6 feet  above  the  datum  plane.  Let  a stake,  marked  6,  be 
driven  here,  and  direct  the  rodman  around  the  hill,  until  a 
second  position  shall  be  found,  when  the  horizontal  wire  of  the 
telescope  will  cut  the  vane,  and  drive  there  another  stake, 
marked  6 ; and  so  on,  until  a sufficient  number  of  stakes  have 
been  driven  to  determine  the  curve  (6).  Then,  let  the  line  of 
stakes,  marked  6,  be  surveyed  with  the  compass  and  chain,  and 
plotted.  Other  contour  lines  may  be  found  in  a similar  manner. 


310 


ELEMENTS  OF  SURVEYING. 


[HOOK  VIII. 


343.  A practical  application  of  this  method  is  also  called  for 
when  a surveyor  is  required  to  determine,  in  advance,  the  area 
of  land  which  will  be  submerged  by  the  construction  of  a dam. 
Iu  this  case  the  contour  plane  is  fixed  by  the  proposed  height  of 
the  dam.  Having  set  up  at  any  point,  a reading  is  taken  upon 
the  rod  held  upon  a point  at  the  proposed  height  of  the  dam,  and 
the  rodman  is  then  directed  to  points  along  the  shore  of  the 
proposed  pond,  or  reservoir,  which  shall  give  the  same  reading. 
In  this  way  the  contour  of  the  pond,  when  there  is  no  overflow  at 
the  dam,  can  be  determined.  If  the  calculated  depth  of  water 
upon  the  dam  at  the  maximum  overflow  be  added  to  the  rod 
reading  used  above,  the  high-water  contour  may  be  also  traced 
out. 

344.  When  the  plane  of  reference  is  so  chosen  that  points 
of  the  work  fall  on  different  sides  of  it,  all  the  references  on  one 
side  are  called  positive,  and  those  on  the  other,  negative.  The 
curves  having  a negative  reference  are  distinguished  by  placing 
the  minus  sign  before  the  number  ; thus  — ( ). 

Shading  and  Delineation. 

345.  Fig.  159  represents  a piece  of  ground  sloping  towards 
A which  is  the  lowest  point ; and  through  this  point  the  plane 
of  reference  is  supposed  to  pass.  The  following  table  indicates 
the  heights  of  the  several  points  above  the  plane  of  reference. 


Ft. 

Ft. 

Ft. 

Ft. 

c above 

D,  2 

H above 

A 7 

p above  A 

9 

B about 

A 

d “ 

A 4 

7c 

A 7 

q “ D, 

9 

L 

(( 

A 13 

h 

A 4 

s 

(( 

A 7 

C “ A 

9 

Cr 

(t 

A 14 

t 

A 4 

f 

a 

A 8 

n “ A 

11 

a 

a 

A 15 

<J  “ 

A 5 

T 

a 

A 8 

i “ A 

12 

F 

<t 

A 15 

l 

A 5 

b 

a 

A 9 

m “ D, 

12 

E 

a 

A 17 

A above  D,  20  feet. 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


311 


The  first  horizontal  plane  is  passed  2 feet  above  D , and  the 
curve  of  intersection  with  the  surface  passes  through  c.  The 
second  secant  plane  is  passed  at  3 feet  above  D , and  intersects  the 


Fig.  159. 


surface,  in  the  curve  uv,  and  also  near  d , which  is  one  foot  above 
the  curve.  All  the  other  secant  planes  are  passed  at  three  feet 
from  each  other;  and,  comparing  the  height  of  each  point  above 
D,  with  the  curves  lying  nearest,  on  either  side,  the  positions  of 
all  the  points,  with  respect  to  the  curves,  and  with  respect  to 
each  other,  are  easily  seen. 

346.  The  manner  of  shading  the  map,  so  as  to  indicate  the 
hills  and  slopes,  consists  in  drawing  the  lines  of  shading  per- 
pendicular to  the  horizontal  curves,  as  already  explained  (Art. 
337).  These  shading  lines  are  drawn  close  together,  when  the 
slope  is  abrupt,  and  further  apart,  as  it  grows  more  gentle. 
Figure  159  indicates  the  method  of  shading. 

347.  In  topographical  surveys,  great  care  should  be  taken  to 
leave  some  permanent  marks , with  their  levels  written  on  them 


312 


ELEMENTS  OF  SURVEYING. 


[BOOK  yiii. 


in  a durable  manner.  For  example,  if  there  are  any  rocks,  let 
one  or  more  of  them  be  smoothed,  and  the  vertical  distance  from 
the  plane  of  reference  marked  thereon ; or  let  the  vertical 
distance  of  a point  on  some  prominent  building  be  ascertained 
and  marked  permanently  on  the  building.  Such  points  should 
also  be  noted  on  the  map,  so  that  a person,  although  un- 
acquainted with  the  ground,  could  by  means  of  the  map  go 
upon  it  and  trace  out  all  the  points,  together  with  their  differ- 
ences of  level. 

348.  Besides  representing  the  contour  of  the  ground,  it  is 
often  necessary  to  make  a map  which  shall  indicate  the  woodland, 
the  marsh,  roads,  ditches,  etc.  For  this,  certain  characters,  or 
conventional  signs,  have  been  agreed  upon,  as  the  representatives 
of  things,  and  when  these  are  once  fixed  in  the  mind,  they 
readily  suggest  the  objects  for  which  they  stand.  Those  which 
are  given  in  the  four  following  pages  have  been  adopted  by  the 
U.  S.  Coast  and  Geodetic  Survey,  and  are  used  in  all  plans  and 
maps  made  under  its  direction. 

It  is  very  desirable  that  a uniform  method  of  delineation 
should  be  adopted,  and  it  is,  therefore,  recommended  that  the 
conventional  signs  given  in  the  accompanying  plates  be  carefully 
studied  and  uniformly  followed. 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


313 


Fig.  100.— Sparsely  settled  Town,  Salt  Marsh,  Pine  Woods,  Bitches,  Fences,  and  Undefined 

Roads. 


Fig.  101. — Blocking  of  Cities,  Large  Buildings,  Suburban  Villas  and  Grounds,  Fresh  Marsh. 


314 


ELEMENTS  OF  SURVEYING. 


[BOOK  VIII. 


Fig.  102. — Railroads,  Canals,  Iron  bridges,  Rocky-cliffs,  Mid-river  drift,  Water-worn  Rocks. 
Mixed  Woods  over  hill  curves. 


BOOK  VIII.] 


TOPOGRAPHICAL  SURVEYING. 


315 


Fig.  164.— Heavy  Oak  Woods,  Reclaimed  Marsh,  and  Orchards. 


Fig.  165.— Fresh  Water  Pond,  Meadow  Grass,  Sage  Brush,  and  Arroyos. 


ELEMENTS  OF  SURVEYING, 


[BOOK  VIII, 


310 


Fig.  166.— Sand  and  Shingle  Beaches,  Eroded  Earth  Banks,  Roads,  Fences,  Shaded  Road- 
sides, Hill-shading. 


BOOK  IX. 

RAILWAY  CURVES. 


349.  After  the  route  of  a railroad  or  canal  has  been  deter- 
mined by  reconnaissance , the  centre  line  of  the  work  is  estab- 
lished by  a transit  (see  Section  III,  B.  IV). 

350.  This  preliminary  survey  establishes  a succession  of 
straight  lines,  of  greater  or  less  length,  according  to  the  obsta- 
cles to  be  avoided  or  the  advantages  to  be  gained,  arising  from 
the  nature  and  the  contour  of  the  ground. 

The  angle  formed,  at  each  change  in  the  direction  of  the 
route,  is  carefully  measured  and  recorded. 

In  the  final  survey  or  location,  these  angles  are  replaced 
by  curves ; and  in  order  that  the  change  in  direction  shall 
be  as  gradual  as  practicable,  the  straight  lines  of  direction 
are  made  tangents  to  the  curves  at  their  point  of  meeting. 

The  preliminary  survey  is  termed,  by  the  engineer,  “running 
out  tangents.” 

351.  We  will  proceed  to  describe  the  method  of  locating 
curves,  first  giving  the  mathematical  principles  applicable  to 
the  subject. 

Let  AD  and  DB , Fig.  168,  be  two  tangents  to  the  arc  of  a 
circle,  AB.  Draw  the  radii  AO,  BC , and  the  secant  CD. 

The  following  relations  are  easily  deduced.  The  tangents 
AD  and  DB  arc  equal  (Leg.,  Bk.  Ill,  Prob.  14).  The  angles 
A and  B are  right  angles  (Leg.,  Bk.  Ill,  Prop.  9),  consequently 
the  angles  C and  D,  of  the  quadrilateral  ADBC , must  be  sup- 


318  ELEMENTS  OF  SURVEYING.  | BOOK  IX. 


plements  of  each  other.  The  angle  TDB , therefore,  must  be 


The  angle  TDB  is  the  angle  formed  by  two  straight  lines 
of  the  preliminary  survey,  and  is  carefully  measured  by  the 
engineer,  in  locating  tangents. 

From  formula  (1),  we  can  determine  the  value  of  d , for 
any  given  values  of  a and  r ; and  hence  we  can  determine 
at  what  point  on  the  tangent,  laid  off  from  D,  the  curve  of 
any  given  radius  must  commence. 

It  is  evident,  both  from  the  diagram  and  the  formula,  that 
for  any  given  angle  between  the  tangents,  the  greater  the 
radius  of  the  curve,  the  greater  will  be  the  distance  cut  off 
between  the  intersection  of  the  two  tangents  and  the  point  of 
tangency. 

It  is  sometimes  necessary  to  give  a particular  value  to  d . 
In  such  case,  we  use  the  formula, 

r = d cot  \a.  (2) 

352.  T1  le  work  of  laying  out  or  locating  a curve  in  the 
field  is  somewhat  simplified,  if  the  curve  have  such  dimensions 
that  one  chain,  of  100  feet,  have  an  arc  corresponding  to  an 
exact  n umber  of  degrees. 


BOOK  IX.] 


RAILWAY  CURVES. 


319 


The  radii  of  such  curves  are  easily  calculated.  Thus,  a 
circle  in  which  one  degree  of  arc  measures  one  chain,  will 
have  a circumference  of  360  chains,  or  of  36,000  feet,  and  con- 
sequently, a radius  of  = 5729.58  feet. 

^ J 2x3.1416 


In  a circle  in  which  two  degrees  of  arc  correspond  to  a 
chain,  the  radius  will  be  only  half  as  great,  or  2864.79. 

When  three  degrees  of  arc  measure  one  chain,  the  radius  is 


5729.58 

3 


= 1909.85  feet. 


The  number  of  degrees,  corresponding  to  one  chain,  of  a 
railway  curve,  is  called  the  “degree  of  curvature” 

The  radius  of  a one-degree  curve  is  5729.58  feet;  of  a two- 
degree  curve,  2864.79  feet,  &c.  ^ 

Representing  the  degree  of  curvature  by  c,  we  have  the 
formula, 

r>  5729.58 

-j~'  () 

r being  expressed  in  feet,  and  c being  the  number  of  degrees 
corresponding  to  100  feet  of  arc,  or  in  common  practice  to 
100  feet  chord. 


353.  Apply  the  preceding  formulas  (1),  (2),  (3),  to  the  fol- 
lowing examples: 


1.  If  the  angle  TUB , of  the  tangents,  he  45°  10',  what  dis* 
tance  must  be  laid  off  from  the  intersection  D , to  the  point  of 
tangency,  to  admit  of  a 4°  curve  ? From  formula  (3),  we  have, 


r = 


5729.58 

\ 


1432.39  feet. 


Substituting  this  value  of  r in  formula  (1),  we  have, 

d = 1432.39  tang  22°  35'  = 595.76  feet. 

2.  If  the  angle Wt  be  30°,  and  the  distance  d be  600  feet, 
what  is  the  radius?  A ns.  2239.2  feet. 


320 


ELEMENTS  OF  SURVEYING.  [BOOK  IX. 


3.  What  is  the  degree  of  curvature  in  the  last  example? 
Formula  (3)  gives 

5729.58 


= 2.°5587  = 2°  33'  31". 


4.  The  angle  a being  20°  21',  what  is  the  value  of  a for 

Arts. 


a one-degree  curve  ? 


Location  of  Curves  by  one  Transit. 


354.  The  location  of  curves,  according  to  the  most  common 
method,  consists  in  laying  off,  at  the  point  of  tangency  A , such 
angles  as  shall  just  subtend  one  chain  of  arc. 

If  the  arcs  Av , vw,  wx,  &c.,  Fig.  169,  represent  arcs  of  one 
chain  each,  the  angles  ACv,  vCw,  &c.,  are  each  equal  to  the 
degree  of  curvature. 


The  angles  DAv , vAw , ivAx,  are  each  equal  to  one-half  the 
degree  of  curvature  (Leg.,  Bk.  Ill,  Prop.  7,  Sell.,  and  Prop.  21). 

The  operations  in  the  field  are  very  simple.  The  party 
should  consist  of  a transitman,  two  chainmen,  and  an  axe-man. 


BOOK  IX.] 


RAILWAY  CURVES. 


321 


The  transit  is  set  and  adjusted  at  a tangent  point,  as  A,  and 
directed  along  the  tangent  toward  D. 

An  angle  equal  to  half  the  degree  of  curvature  is  deflected 
from  AD  toward  the  side  on  which  the  curve  is  to  run.  The 
hind-chainman  holds  his  end  of  the  chain  at  A.  The  fore- 
chainman,  keeping  the  chain  carefully  extended,  is  directed  by 
the  transitman  into  line  with  the  axis  of  the  telescope.  This 
locates  the  point  v on  the  curve. 

From  the  line  Av,  another  deflection  is  now  made,  of  the 
same  angle  as  before.  The  chainmen  move  forward  ; the  hind- 
chainman  stopping  at  v,  while  the  fore-chain  man,  keeping  the 
chain  extended,  is  directed  by  the  transitman  as  before,  and  a 
second  stake,  w,  is  fixed  on  the  curve.* 

By  continuing  the  process  of  deflecting  angles  equal  to  half 
the  degree  of  curvature,  and  causing  these  angles  to  subtend 
measured  distances  of  one  chain  each,  the  entire  curve  is 
located. 

355.  The  last  deflection  on  the  curve  rarely  corresponds  to 
an  entire  chain  ; it  is,  therefore,  less  than  the  others.  Its  amount 
can  be  readily  calculated,  when  it  is  remembered  that  the  sum 
of  all  the  deflections,  or  the  angle  DAB,  is  exactly  equal  to 
one-half  the  angle  a. 

Hence,  if  from  ~ the  sum  of  the  deflection  angles  laid  off 

be  subtracted,  the  remainder  will  be  the  final  deflection  angle, 
called  the  sub- deflection  angle.  The  corresponding  sub-cliord  is 
such  part  of  100  ft.  as  the  sub-deflection  angle  is  of  the  deflectio7i 
angle.  In  practice  the  sub-chord  should  always  be  laid  off,  using 
the  sub-deflection  angle,  to  check  the  work  ; the  final  peg,  thus 
located,  should  fall  upon  the  previously  determined  tangent-peg. 

By  this  method  of  laying  out  curves,  any  error  in  chaining 
any  one  chord  is  carried  into  all  succeeding  portions  of  the  curve. 


ELEMENTS  OF  SURVEYING. 


[BOOK  IX. 


322 


356.  It  is  sometimes  necessary  to  remove  the  transit  from 
the  transit- point  to  some  other  point  on  the  curve,  before  the 
location  has  been  completed. 

In  sucli  a case,  the  direction  of  the  tangent  to  this  new  point 
should  be  determined.  Suppose  x,  Fig.  109,  to  be  a located  point 
on  the  curve  to  which  the  transit  has  been  transferred,  and 
from  which  new  points  beyond  x are  to  be  located.  Adjust  the 
transit  and  direct  the  telescope  to  A.  Lay  off  the  angle  Axt , 
equal  to  DAx  (the  sum  of  the  deflections  made  in  locating 
v , w,  and  x ), — xt  is  the  tangent.  By  revolving  the  telescope, 
the  tangent  is  produced  to  s,  from  which  deflections  may  be 
made  as  at  first. 

Note  1. — The  selection  of  the  radius  is  governed  by  cir- 
cumstances. Curves  of  the  longest  possible  radius  are,  in  rail- 
roads, always  the  most  desirable ; but  the  larger  the  radius  for 
any  particular  pair  of  tangents,  the  greater  the  distance  by 
which  the  curve  will  depart  from  the  intersection  of  the 
tangents.  It  may  happen,  therefore,  that  too  large  a radius 
may  lead  to  an  obstacle,  which  the  angle  in  the  first  survey 
was  made  to  avoid. 

The  map,  therefore,  of  the  preliminary  survey,  should  include 
so  much  of  the  topography  of  the  adjacent  section,  that  the 
radii  of  the  curves  may  be  selected  by  an  inspection  of  the 
map. 

Note  2. — It  will  be  observed  that  it  is  the  chord,  and  not 
the  arc,  that  is  measured  for  each  deflection,  when  locating 
in  the  field  ; the  difference,  in  railway  curves,  of  proper  dimen- 
sions, does  not  lead  to  sensible  error. 

For  curves  of  a short  radius  (less  than  two  thousand  feet), 
the  error  may  be  diminished,  by  locating  the  stakes  at  half- 
chain distances,  deflecting,  of  course,  half  the  calculated  deflec- 
tion angle. 


BOOK  IX.  J 


RAILWAY  CURVES. 


323 


Location  of  Curves  by  two  Transits. 

357.  The  surface,  over  which  it  is  necessary  to  locate  a curve, 
may  be  of  such  a character  as  to  render  it  impracticable  for  the 
chain  men  to  make  their  measurements ; if,  however,  the  various 
points  are  accessible  to  the  axeman,  as  in  the  case  of  marshes, 
shallow  lakes,  or  bays,  the  stakes  may  be  accurately  located  by 
the  simultaneous  deflections  of  two  transits. 

The  method  is  based  on  the  following  geometrical  principle  : 


Let  A and  B be  the  two  tangent  points  of  the  curve  AvB 3 
and  D the  intersection  of  the  tangents. 

If  from  any  point  v , on  the  curve,  the  lines  vA,  vB,  be  drawn, 
then  the  sum  of  the  angles  v A B and  vBA  is  measured  by  one- 
half  the  arc  AB,  and  is  therefore  equal  to  one-half  the  angle  a, 
or  to  either  of  the  entire  angles  A or  B. 

To  locate  the  curve  in  the  field,  a transit  is  set  at  each  of 
the  tangent  points  A and  B,  and  the  deflection  angle  is  deter- 
mined as  in  the  first  method. 

The  transitman  at  A deflects,  in  the  usual  way,  one  deflec- 
tion angle  from  the  tangent  AD.  At  the  same  time,  the  transit- 
man  at  B deflects  the  same  angle  from  the  chord  BA,  or  what 
amounts  to  the  same,  he  deflects  the  difference  between  this 
angle  and  \a,  from  the  tangent  BD.  The  lines  of  sight  of  the 


324 


ELEMENTS  OF  SURVEYING. 


[ROOK  IX. 


two  telescopes  now  intersect  at  a point  v,  on  the  curve,  one  chain 
from  A.  The  flagman,  directed  at  the  same  time  by  both 
transitmen,  is  readily  brought  to  the  location  of  the  point. 
By  a repetition  of  this  process  the  entire  curve  is  located. 

Location  of  Curves  by  the  Chain  alone. 

358.  It  is  sometimes  convenient  to  locate  a curve  without 
using  the  transit.  In  such  case,  the  following  method  is  gen- 
erally employed. 

Let  A represent  the  point  of  tangency,  C the  centre,  and 
v , w,  x , located  points  of  the  curve,  one  chain  apart. 

From  v,  draw  vu  perpendicular  to  the  tangent,  and  it  will 
be  the  first  offset,  which  denote  by  o.  Denote  the  length  of  the 
chain  by  l,  and  the  radius  AC,  by  r.  If,  now,  we  suppose  AC 
to  be  prolonged  till  it  meets  the  circumfeience  in  some  point, 
on  the  other  side  of  the  centre  C,  and  this  point  then  to  be 


joinecr  with  v,  and  vn  then  drawn  parallel  to  the  tangent,  we 
shall  have  (Leg.,  Bk.  IV,  P.  23), 

p 

Av2  = 2 r • An  ; hence,  o = — • (4) 

If,  now,  we  prolong  Av  till  vt  = Av,  and  join  t and  w , 


JBOOK  IX.] 


RAILWAY  CURVES. 


325 


tw  will  be  the  second  offset,  and  will  be  double  vu.  For,  the 
triangles  in  the  figure,  whose  vertices  are  C,  and  whose  bases 
are  the  equal  chords  Av,  vw , &c.,  are  isosceles  and  equal. 
Now,  in  any  one  of  the  triangles,  the  sum  of  the  two  angles 
at  the  base  and  the  vertical  angle  C,  is  equal  to  two  right- 
angles.  But,  since  Avt  is  a straight  line,  tvw  + wv G 4-  CvA , 
is  also  equal  to  two  right  angles.  Therefore,  tviv  is  equal  to 
any  one  of  the  equal  angles  at  C,  and  is,  consequently,  double 
the  angle  uAv,  which  is  half  the  angle  C. 

Since  the  triangle  wvt  is  isosceles,  if  vp  be  drawn  perpen- 
dicular to  the  base,  it  will  bisect  both  the  base  and  the  vertical 
angle,  making  tp  = pw.  But  the  right-angled  triangles  Auv  and 
vtp  are"  equal  (Leg.,  Bk.  I,  P.  6);  hence,  tw  = 2 vu.  Denoting 
the  second  offset  by  o',  we  have, 


359.  The  practice  is  as  follows : Having  calculated  the  first 
offset  0,  fix  one  end  of  the  chain  at  A,  Fig.  171,  and  stretch  the 
chain  to  v , measuring  uv  = 0 perpendicular  to  the  tangent ; the 
measurement  of  0 may  be  made  with  a short  supplementary  steel 
tape,  its  end  division  being  marked  to  hundredths  of  a foot,  or 
with  a rod  marked  with  two  divisions  each  equal  to  0.  Next 
hold  one  end  of  the  chain  at  v and  align  a pin  at  t,  on  Av  pro- 
longed, making  vt  = Av ; then  swing  the  chain  towards  w till 
the  measured  distance  tw  = 2o  = o'.  Prolong  vw  as  before  and 
find  a new  point  x,  and  so  continue  till  all  the  full  chords  have 
been  laid  down.  The  last  chord  is  usually  a sub-chord  and  its 
offset  must  be  calculated  separately.  Suppose  it  is  required  to 
run  a curve  of  1000  feet  radius,  the  angle  of  intersection,  a , Fig. 

170,  being  48°.  From  the  formula  d = r tan  we  find  DA  = 

445.2,  and  then  locate  the  tangent  peg  at  A,  and  also  that  at  B. 


32G 


ELEMENTS  OF  SURVEYING. 


[BOOK  IX. 


We  also  find  o = 5 and  o'=  10.  The  angle  at  the  centre  sub- 
tended by  the  100-ft.  chain  is  found  by  the  relation  — = 

r 

sine  \ angle  at  centre  (Leg.,  Trig.,  Art.  64).  In  the  case  supposed 
we  find  the  angle  subtended  to  be  5°  44'.  The  number  of  full 

48° 

chords  is  equal  to  ^0^9  which  gives  8 full  chords,  and  a remain- 
ing angle  at  the  centre  of  2°  8'.  To  find  the  sub-chord  we  have 

sine  ('  ~ ) x 1000  x 2 = 37.24  feet. 

The  final  offset  is  — 1.39, 

r 

The  peg  at  the  end  of  the  sub-chord  should  fall  at  the  second 
tangent  peg,  if  no  errors  have  been  made  in  chaining. 

Note. — In  employing  this  method  of  locating  curves,  the 
aligning  by  which  the  chords  are  produced  should  be  done  with 
much  care,  as  any  error  in  locating  a stake  involves  much 
greater  and  increasing  errors  in  succeeding  stakes. 

This  is  called,  by  engineers,  “ the  method  by  offsetting  from 
tangent  and  chords  produced.” 

EXAMPLES. 

1.  What  are  the  tangent  and  chord  offsets,  for  a curve  of 
2000  feet  radius  ; the  stakes  to  be  100  feet  apart  ? 

Ans.  From  tangent,  2.5  ft.;  from  chord  produced,  5 ft. 

2.  Find  offsets  for  a one-degree  curve. 

Ann.  Tangent,  .87  ft.;  chord,  1.74  ft. 

360.  Let  it  be  required  to  run  out  a curve  of  500  ft. 
radius,  the  stations  being  25  feet  apart  on  the  curve. 

As  the  chord  of  the  angle  i is  to  be  25  ft.,  we  have  (Fig.  172), 


BOOK  IX.]  RAILWAY  CURVES.  327 

. i 12i 

sine2  = 5oo  = -025 

from  which  we  find  i = 2°  52\  Now, 

Tx  — bs  = sine  i x 500  = 25  feet ; 
and  xb=  Ts  = 500— (cos  i x 500)  = .63  ft. 


Measure  from  T 25  feet  to 
x,  and  at  x lay  off  the  per- 
pendicular xb  = .63  ft.,  thus 
locating  the  point  b on  the 
curve.  To  locate  c we  have 

Ty  — wc  = sin  2 i x 500 
= 49.95  ft.,  and 

yc  — Tw  500  — (cos  2 i x 500) 

= 2.50  ft. 


The  next  tangent  distance  to  z equals  sine  3 i x 500,  and  the 
offset  at  z equals  500— (cos  3&'x500),  and  so  on  for  succeeding 
points. 

When  the  offsets  become  too  long  to  be  readily  and  correctly 
set  out,  a new  tangent  is  located  thus:  Prolong  cd  to  li  making 
dli  — cd,  and  then  bisect  he  (e  having  been  already  located)  in  f 
and  range  a line  df,  which  line  will  be  tangent  to  the  curve  at  d. 
The  correctness  of  the  new  tangent  should  always  be  checked  by 
locating  from  it  the  third  or  fourth  station  counting  back 
towards  T.  The  same  computed  values  that  were  used  in 
locating  the  points  already  fixed  may  be  again  used  in  locating 
new  points  from  the  new  tangent. 


Laying  off  the  Ordinates. 


361.  The  methods  described  thus  far  for  locating  railway 
curves,  apply  to  points  100  feet  apart.  This  is  sufficiently 


328 


ELEMENTS  OF  SURVEYING. 


[ROOK  IX. 


accurate  for  the  earthwork.  In  laying  the  track,  however,  stakes 
every  ten  or  twelve  feet  are  necessary.  These  are  set  by  drawing 
the  chain  or  tape  in  a straight  line  between  the  100-ft.  stakes, 
and  measuring  from  it,  offsets,  as  often  as  desirable,  to  the 
intermediate  points  of  the  curve. 

The  length  of  these  offsets,  or  ordinates,  is  calculated  in 
the  following  manner : 


Let  VW  represent  a 100-ft.  chord  of  a railway  curve,  of  which 
C is  the  centre.  Draw  the  diameter  HK  parallel  to  VW,  and 
drop  the  perpendicular  VL.  Then, 

VL2  = HL  x LK. 

(Legendre,  Bk.  IV.,  Prop.  23,  cor.  2).  Since 
HL  — r — 50, 
and  LK  = r -f  50, 

the  value  of  VL  is  readily  calculated  for  known  values  of  the 
radius.  . 

Let  NM  be  an  ordinate,  at  any  distance  from  VL,  say 
10  feet.  Then, 

NM2  = IIM  x MIC; 

whence,  NM 2 = (r  — 40)  (r  4-  40). 

Having  determined  NM,  subtract  VL  from  it,  and  we  have 
Nt,  one  of  the  ordinates  required. 


BOOK  IX.] 


RAILWAY  CURVES. 


329 


In  this  manner,  by  calculating  the  full  ordinate  to  the 
diameter,  and  subtracting  VL , any  desired  number  of  offsets  are 
determined  for  the  half  chain  VF.  For  FW,  the  ordinates  have 
the  same  length,  but  are  located  in  the  inverse  order. 

The  middle  ordinate,  FE,  is  found  by  subtracting  VL  from 
the  radius. 


EXAMPLE. 


Determine  the  ordinates  10  feet  apart  on  a 100-foot  chord, 


for  a two-degree  curve.  Radius,  2864.79  feet. 

A ns.  At  10  feet 
At  20  “ 

At  30  “ 

At  40  “ 


= .15  feet. 
= .28  “ 
= .36  “ 
= .42  “ 


Middle  ordinate  = .43  “ 


Reversed  and  Compound  Curves.* 

362.  Two  curves,  of  the  same  or  different  radii,  may  join 
each  other  and  have  a common  tangent  at  the  point  of  junction 
If  the  curves  lie  on  opposite  sides  of  the  common  tangent,  they 
form  a reversed  curve,  and  their  radii  may  be  equal  or  unequal ; 
if  they  lie  on  the  same  side  of  the  common  tangent,  they  have 
unequal  radii  and  form  a compound  curve.  Thus  ABC  (Fig.  174) 
is  a reversed  curve,  and 
ABD  a compound  curve. 

The  point  of  contact 
of  the  common  tangent 
is  called,  in  a reversed 
curve,  the  reversing  point, 
and,  in  a compound 
curve,  the  common  tan- 
gent point  or  the  point 

of  compound  curvature.  • Fia.  m. 


* Taken,  by  permission  and  with  slight  alterations,  from  Ilenck’s  “ Field  Book  for 
Railroad  Engineers.” 


$30  ELEMENTS  OF  SURVEYING.  [ROOK  IX. 


The  reversing  point  of  a reversed  curve  contained  be- 
tween parallel  tan- 
gents is  in  the  line 
joining  the  tangent 
points.  Thus,  in  the 
curve  ACB,  Fig.  175, 
contained  between  the 
parallel  tangents  HA 
and  BK,  the  reversing 

® Fio.  175. 

point,  Cy  is  in  the  line 

AB  joining  the  points  of  contact  A and  B . 

A reversed  curve  is  of  use  in  certain  track-work  near  stations, 
constructing  turnouts,  &c.,  though  it  is  not,  or  ought  not  to  be, 
used  on  the  ordinary  running  track. 


363.  Given  the  perpendicular  distance  betiveen  two  parallel 
tangents  BD  = h {Fig.  175),  the  distance  betiveen  the  two  tangent 
points  AB  = a,  and  the  first  radius , EC  = R,  of  a reversed 
curve  uniting  the  tangents  HA  and  BK,  to  find  the  chords 
AC  — a’  and  CB  = a",  and  the  second  radius  CF  = R'. 


Draw  the  perpendiculars  EG  and  FL  (Fig.  1 75).  Then  the 
right-angled  triangles  ABD  and  EAG  have  the  angle  BAD 
= AEG,  since  each  is  one-half  AEG,  and  are,  therefore, 
similar ; hence, 

AB  : BD  ::  EA  : AG, 


or 


Since 


a : b : : R : 

, 2Rb 

. * . a = • 

a 

a " + a'  = a, 
a"  = a — a! . 


a) 

(2) 


To  find  R',  we  have,  from  the  similar  triangles  ABD  and  FBL, 
a : b : : IV  : \a". 


BOOK  IX.] 


RAILWAY  CURVES. 


331 


R'  = 


aa 

~2b 


(3) 


Any  three  of  the  quantities  a,  a ',  a",  b,  R,  R',  being  given,  the 
others  may  be  found  from  equations  (1),  (2),  (3). 


EXAMPLES. 

1.  b = 8,  a = 160,  R = 900 ; find  a',  a",  R\ 

a'  = 90 ; a"  = 70;  R'  = 700. 

2.  b = 8,  a'  = 90,  a"  = 70  ; find  a,  R,  R\ 

3.  R = 900,  R'  = 700,  b — 8 ; find  a,  a ',  a". 


364.  Given  the  line  AB  — a (Fig.  176),  which  joins  the  fixed 
tangent  points  A and  B,  the  angle  DAB  = A,  and  the  angle 
ABG  = B,  to  find  the  common  radius  EC  — CF  — R of  a 
reversed  curve  to  unite  the  tangents  HA  and  BL. 


From  the  triangle 
A EE,  Fig.  176,  we 
have  (Art.  32), 

AE  . EE  : : sin  A EE 
: sin  EAE ; 

but  since 

EAE  = 90°  — A, 
sin  EAE  = cos  A; 


Fig.  176. 


hence,  denoting  angle  A EE  by  E,  we  have, 

EE  • sin  E=  R • cos  A.  (1) 

In  like  manner,  from  the  triangle  BFE , we  have, 

FE  • sin  E = R • cos  B.  (2) 


From  (1)  and  (2),  by  addition,  we  have, 

(EE+  FE)  sin  E = 2R  sin  E = R (cos  A +cos  B).  (3) 


332 


ELEMENTS  OF  SURVEYING. 


[KOOK  IX. 


. * . sin  AT  = J (cos  A + cos  A).  (4) 

But  cos  A +cos  B = 2 cos  \ ( A + B)  cos  | (A  — B), 

(see  Legendre,  Trig.  Art  67) ; hence, 

sin  K = cos  J (A  -J- A)  cos  \ (A—B).  (5) 

Having  found  K , we  have  the  angle 

ABIC  = E = 180°— (AT+ AAAT)  = 180o-(AT+90°-A) 

= 90°  + A — AT.  (6) 

In  like  manner,  the  angle 

BFIC  = F = 90°  + A— A.  (7) 


From  the  triangle  AEK , we  have 

AE  : AK  : : sin  K : sin  E. 

. * . A?  sin  E = AAf  sin  AT.  (8) 

In  like  manner,  from  triangle  BFK , we  have, 

A?  sin  F = BK  sin  A.  (9) 

From  (8)  and  (9),  by  addition,  we  have 

A (sin  A' -f  sin  A7)  = (AA+  AA^)  sin  A7”  = a sin  K.  (10) 
But,  sin  A-fsin  F=  2 sin  \ (A+A)  cos  £ (A— A), 

(see  Legendre,  Trig.  Art.  67) ; hence, 

2 R sin  \ (A+ A)  cos  \ (E—F)  = a sin  K. 

. R - Sln  K (n) 

'•  sin  £ (E+F)  cos  * (E-F)  K ’ 

Substituting  in  (11)  the  values  of  A and  A from  (G)  and  (7), 
we  have 


n } a sin  K 

K ~ cos ~[K-l  P + 2?j]  cos  i (A—B)' 


(12) 


Example. — Given  a = 1500,  A = 18°,  B = 6°;  find  R. 
From  equation  (4),  K is  found  to  be  76°  3G'  10",  and  then 
from  (12),  R is  found  to  be  1710.48. 


BOOK  IX.  J 


RAILWAY  CURVES. 


333 


365.  If  one  branch  of  a compound  curve  be  produced  until 
the  tangent  at  its  extremity  is  parallel  to  the  tangent  at  the 
extremity  of  the  second  branch , the  common  tangent  point  of  the 
two  arcs  is  in  the  straight  line  produced  which  passes  through 
the  tangent  points  of  these  parallel  tangents. 

Let  AGB , Fig.  177, 
be  a compound  curve 
uniting  the  tangents  HA 
and  BK.  The  radii  EG 
and  FG,  being  perpen- 
dicular to  the  common 
tangent  at  G,  the  point 
of  compound  curvature, 
are  in  the  same  straight 
line.  Continue  the  curve 
AG  to  D,  where  its  tan-  fig.  m. 

gent  OD  becomes  parallel  to  BK,  and  consequently  the  radius 
ED  parallel  to  FB.  Then  if  the  chords  GD  and  CB  be 
drawn,  we  have  the  angle  CED  = CFB,  whence  EGD,  the  half- 
supplement of  GED,  is  equal  to  FGB,  the  half-supplement  of 
GFB.  But  EGD  can  not  be  equal  to  FGB,  unless  CD  coincides 
with  CB;  therefore,  the  line  BD  produced  passes  through  the 
common  tangent  point  G. 

366.  To  find  a limit  in  one  direction  of  each  radius  of  a 
compound  curve. 

Let  AI  and  BI,  Fig.  177,  be  the  tangents  of  the  curve  ; draw 
IM  bisecting  the  angle  AIB  ; draw  AL  and  BM  perpendicular 
respectively  to  A I and  BI,  meeting  IM  in  L and  M.  Then  the 
radius  of  the  branch  commencing  on  the  shorter  tangent,  A I, 
must  be  less  than  AL,  and  the  radius  of  the  branch  commencing 
on  the  longer  tangent,  BI,  must  be  greater  than  BM. 


ELEMENTS  OF  SURVEYING. 


[BOOK  IX. 


334 


For,  suppose  the  shorter  radius  equal  to  AL,  and  hence,  IN 
equal  to  Al\  join  LN,  then  the  equal  triangles  AIL  and  NIL 
give  AL  = LN ; so  that  the  curve,  if  continued,  will  pass 
through  N,  where  its  tangent  will  coincide  with  IN.  Then 
(Art.  365)  the  common  tangent  point  would  be  the  intersection 
of  the  straight  line  through  B and  N with  the  first  curve  ; but 
in  this  case  there  can  be  no  intersection,  and  therefore  no  com- 
mon tangent  point. 

Suppose,  next,  that  this  shorter  radius  is  greater  than  AL , 
and  continue  the  curve  till  its  tangent  becomes  parallel  to  BI. 
In  this  case,  the  extremity  of  the  curve  will  fall  outside  the  tan- 
gent BI  in  the  line  AN  produced,  and  a straight  line  through 
B and  this  extremity  will  again  fail  to  intersect  the  curve  already 
drawn.  As  no  common  tangent  point  can  be  found  when  the 
shorter  radius  is  taken  equal  to  AL , or  greater  than  AL,  no 
compound  curve  is  possible.  This  radius  must,  therefore,  be  less 
than  AL. 

In  like  manner  it  may  be  shown  that  the  radius  of  the  other 
branch  of  the  curve  must  be  greater  than  BM. 

If  the  tangents  A I and  BI,  and  the  intersection  angle  I are 
known,  then 

AL  = AI  tan  \I\ 

BM  — BI  tan  J/. 

These  values  are,  therefore,  the  limits  of  the  radii  in  one 
direction. 

367.  If  nothing  were  given  but  the  position  of  the  tangents 
and  the  tangent  points,  it  is  evident  that  an  indefinite  number 
of  different  compound  curves  might  connect  the  tangent  points  ; 
for  the  shorter  radius  might  be  taken  of  any  length  less  than  the 
limit  found  above,  and  a corresponding  value  for  the  greater 
could  he  found.  Some  other  condition  must,  therefore,  be  intro- 
duced, as,  for  example,  in  the  following  problem: 


BOOK  IX.] 


RAILWAY  CURVES. 


335 


368.  Given  the  line  AB  = a (Fig.  177),  which  joins  the  jixed 
tangent  points  A and  B,  the  angle  BAI  — A,  the  angle  ABI—  B, 
and  the  first  radius  AE  = R,  to  find  the  second  radius  BF  = R' 
of  a compound  curve  to  unite  the  taiigents  HA  and  BE. 

Let  the  first  curve  be  run  with  the  given  radius  from  A to  D, 
where  its  tangent  DO  becomes  parallel  to  BI ; then  the  common 
tangent  point  C is  in  the  line  BD  produced,  and  the  chord 
CB=  CD  + BD. 

The  angle  OAD , formed  by  a tangent  and  a chord,  is 
measured  by  half  the  arc  A CD ; hence,  R sin  OAD  = R sin 
I AD  = \AD  (see  Legendre,  Trig.  Arts.  64  and  30) ; hence, 

AD  — 2 R sin  IAD.  (1) 

In  the  triangle  OAD,  since  OA  and  OD  are  equal,  angle  AOD  = 
180° — %OAD\  hence,  OAD=IAD=W°—\AOD=W°—\AIB; 
but  from  the  triangle  AIB , \AIB  — 90°— £ (A  +B) ; 

whence  the  angle  IAD  = \ (A  -f-  B).  (2) 

From  (1)  and  (2),  we  have 

AD  = 2R  sin  J (A  +B).  (3) 

Then  in  the  triangle  ABD,  we  have  AB  = a,  AD  = 2 R sin 
\ ( A + B ),  and  the  included  angle 

DAB  = I A B— IAD  = A— | (A  + B)  = * (A—B)  ; 

whence,  we  have  the  proportion  (Legendre,  Trig.  Art.  45), 

AB  + AD  : AB-AD  ::  tan  %(ADB+ABD) 

: tan  \{ADB-ABD).  (4) 

\ (ADB  + ABD)  = \ (180° — DA B)  ; \ ( ADB-ABD ) may  be 
found  from  (4);  and  from  the  half-sum  and  half-difference  thus 
obtained,  the  angles  ADB  and  ABD  may  be  found.  These 


ELEMENTS  OF  SURVEYING. 


336 


[book  IX. 


angles  being  known  the  side  BD  may  be  found  from  the 
proportion  (Art.  32), 

sin  ABB  : sin  DAB  ::  AD  : BD. 


The  angle  CBI  = B-ABD. 

Again,  as  above,  CB  = 272'  sin  CBI , 

and  CD  = 2 R sin  CDO  = 272  sin  CBI. 


Substituting  these  values  of  CB  and  CD,  in  the  equation 
CB  = CD  + BD, 


we  have 

272'  sin  CBI  = 272  sin  CBI  A-  BD. 

7“>/  D /2/) 

R ~ R + 2 sin  CBI ’ 

(5) 

or 

8 <*-*)  = »T„W 

(6) 

When  the  angle  B is  greater  than  A,  that  is,  when  the  greater 
radius  is  given,  the  solution  is  the  same  except  that  the  angle 
DAB  — ^ ( B—A ),  and  CBI  is  found  by  subtracting  the  supple- 
ment of  ABD  from  B.  We  shall  also  find  CB  = CD—BD;  hence, 


R'  = R 


BD 

2 sin  CBV 


(7) 


or 


2(72-72')  = 


BD_ 
sin  CBI 


Note. — If  more  convenient,  the  point  D may  be  determined 
in  the  field  by  laying  off  the  angle  1AD  — £ (A  + B),  and 
measuring  the  distance  AD  = 2 R sin  £ (A  4-7?).  BD  and  CBI 
may  then  be  measured,  instead  of  being  calculated  as  above. 

Example,  a = 950,  A — 8°,  B = 7°,  72  = 3000;  find  72'. 

AD  = 2x3000  sin  $ (8° + 7°)  = 783.16 


BOOK  IX.]  RAILWAY  CURVES. 

and  DAB  = (8°— 7°)  =30'. 

Then,  to  find  ABD,  we  have 

log  (AB—AD),  166.84  = 2.222300 

log  tan  i (ADB  + ABD),  89°  45'  = 2.360180 

(a.  c.)  log  ( AB  + AD ),  1733.16  = 6.761161 

log  tan  i (ABB— ABD),  87°  24'  17"  = 1.343641 

.-.  ABD  = 2°  20'  43"  ; 

whence,  CBI  = B—ABD  = 4°  39'  17". 


Next,  to  find  BD , 


log  (783.16) 

= 2.893849 

log  sin  DAB  (30') 

= 7.940842 

(a.  c.)  log  sin  (2°  20'  43") 

= 1.388052 

log  BD  (167.01) 

= 2.222743 

Then,  from  (6),  we  have 

log  BD  (167.01) 

= 2.222743 

(a.  c.)  log  CBI  (4°  39'  17") 

= 1.090708 

log  2 (R'—R),  2058.03 

= 3.313451 

£'-72  = 1029.01; 

= 3000  + 1029.01  = 4029.01. 


337 


BOOK  X. 

MINING  SURVEYING. 


SECTION  I. 

DEFINITIONS  AND  GENERAL  PRINCIPLES. 

369.  Mining  Surveying  comprises  all  the  operations  neces- 
sary to  determine  the  relative  positions  of  the  parts  of  a mine 
with  respect  to  each  other,  and  also  with  respect  to  the  surface 
of  the  earth. 

370.  The  general  principles  involved  in  this  branch  of  sur- 
veying are  the  same  as  those  used  in  surface  surveying,  but, 
from  the  nature  of  the  case,  certain  modifications  are  required. 

Stations  are  designated  by  lamps  instead  of  flags,  and  lamp- 
stands  instead  of  flag-rods,  or  by  plumb-lines  properly  suspended 
and  lighted ; station  points , if  temporary,  are  marked  by  cross- 
lines  chipped  in  the  rock,  or  sometimes  by  simple  chalk  lines, 
and,  if  permanent,  by  iron  pegs  driven  into  holes  drilled  for  the 
purpose. 

The  compass  is  rarely  used  for  underground  work,  and 
ought  never  to  be  used  for  any  but  rough  work,  because  of  the 
inaccuracies  to  which  it  may  lead.  A great  deal  of  the  mining 
in  the  U.  S.  is  done  in  the  far  West,  where  the  magnetic  declina- 
tion is  not  known  ; and,  further,  the  declination  is  seriously 
affected  by  the  state  of  the  atmosphere,  the  presence  of  iron  ores, 
magnetic  pyrites,  &c. 

The  transit , which  is  the  principal  angular  instrument  em- 


SEC.  I.] 


DEFINITIONS. 


339 


ployed,  and  the  only  necessary  one,  differs  from  the  ordinary 
transit  in  having  a diagonal  eye-piece,  to  permit  observations  to 
be  made  when  the  telescope  is  directed  vertically  upward,  and 
also  an  arrangement  for  illuminating  the  cross-wires. 

A method  of  lighting  the  cross- wires  is  by  the  reflector  shown 
in  Fig.  178.  This  is  an  elliptical  piece  of  brass, 
silver-plated  on  the  under  side,  and  inclined  at 
an  angle  of  45°  to  its  ring,  which  is  fitted  to  the 
object  end  of  the  telescope;  the  hole  in  the 
reflector  admits  the  use  of  the  telescope,  while 
a light  held  near  the  under  surface  illuminates 
the  cross- wires. 

The  transit  should  be  furnished  with  a solar  attachment  (see 
Appendix  A)  for  establishing  the  true  meridian  and  running  out 
lines  with  reference  thereto ; and  should  also  have  an  extension 
tripod  to  adapt  it  for  use  in  mountain  surveys,  where  one  or 
more  legs  must  be  shortened,  and  for  mines,  where  in  many 
places  a short  tripod  is  indispensable. 

371.  Traversing  is  the  operation  of  running  a zigzag  line, 
from  one  point  to  another.  The  elements  of  the  traverse  are 
straight  lines,  determined  by  their  lengths  and  by  their  in- 
clinations to  certain  fixed  planes.  In  mining  surveying,  three 
such  planes  are  used ; the  first , is  either  a meridian  plane 
through  the  origin  of  the  traverse,  or  a vertical  plane  through 
the  first  course;  the  second , is  a horizontal  plane  through  the 
origin  ; and  the  third , is  a vertical  plane  through  the  origin, 
and  perpendicular  to  the  other  two. 

372.  Working,  or  Reducing  the  Traverse,  is  the  opera- 
tion of  finding  the  length  and  direction  of  a single  line, 
equivalent  to  the  zigzag,  that  is,  starting  from,  and  terminating 
at,  the  same  points.  Such  a line  is  called  the  resultant  of  the 
traverse. 


Fig.  178. 


340 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


The  zigzag  line  is  run  along  the  subterranean  openings  of 
a mine.  Such  openings,  when  vertical,  are  called  shafts , 
and  when  not  vertical,  tunnels. 


SECTION  II. 

METHOD  OF  LOCATING  CLAIMS. 

373.  The  methods  employed  in  locating  claims  on  the  surface 
are  various,  and  in  general  very  crude,  except  when  a surveyor, 
usually  a U.  S.  Deputy  Mineral  Surveyor,  is  called  upon  to  make 
the  survey.  Prospectors  are  usually  without  suitable  instruments 
to  lay  off  their  claims  with  any  degree  of  accuracy.  We  will, 


Fig.  179. 


SEC.  II.] 


METHOD  OF  LOCATING  CLAIMS. 


341 


therefore,  only  consider  the  method  employed  by  the  Deputy 
Mineral  Surveyors,  who  are  compelled  to  follow  the  instructions 
received  from  the  Land-office. 

374.  The  mining  claim  shown  in  Fig.  179  has  the  dimensions 
allowed  by  the  U.  S.  Mineral  Laws,  viz.,  1500  feet  in  the  direc- 
tion, or  on  the  “strike,”  of  the  vein,  by  300  feet  on  each  side  of 
the  middle  of  the  vein  at  the  surface.  An  ideal  case  is  assumed, 
the  exact  strike  is  supposed  to  have  been  determined,  and  the 
side  lines  of  the  claim  run  parallel  to  it.  Such  cases  are  the 
exception.  Generally,  the  strike  is  not  accurately  determined 
by  the  prospector,  and  claims  are,  in  both  direction  and  dimen- 
sions, usually  but  a rough  approximation.  The  courts  are  lenient 
and  allow  considerable  latitude  in  the  matter.  The  prospector 
may  alter  the  boundaries  of,  or  “swing,”  his  claim  after  the  strike 
has  been  determined,  provided  that  no  trespass  is  committed  ^ 
but  when  the  course  cannot  be  altered  without  interference 
with  other  claims,  the  boundaries  already  established  must  be 
adhered  to. 

When  a claim  is  made  and  staked  out,  a record  of  the  same 
must  be  placed  on  file  in  the  office  of  the  Eecorder  having  charge 
of  the  records  of  mining  locations  in  the  district  where  the  claim 
is  situate  ; and  in  case  the  boundaries  of  such  claim  are  altered, 
a record  of  the  amended  location  must  be  filed  in  like 
manner. 

“A  failure  to  make  and  record  the  location  in  accordance 
with  the  law  and  regulation  in  force  at  the  date  of  the  location, 
will  defeat  the  claim  ; and  if  it  is  not  made  with  such  definite- 
ness as  to  operate  as  notice  to  all  persons  seeking  to  acquire 
rights  to  mining  lands,  it  will  be  void  for  uncertainty  ” (General 
Land  Office  Instructions). 

375.  When  mineral  discoveries  are  made  and  located  on  sur- 
veyed land,  the  surveys  must  conform  to  the  public  survey  and 


342 


ELEMENTS  OF  SURVEYING. 


[BOOK  X, 


be  connected  with  or  “ tied”  to  it,  so  that  there  will  he  no  diffi- 
culty in  finding  some  established  corner.  In  Fig.  179,  the  points 
1 and  2 are  tied  to  the  N.  W.  cor.  of  the  quarter-section  in 
T.  9 N.  R.  5 W. 

376.  When  discoveries  are  made  on  unsurveyed  land,  U.  S. 
locating  monuments  are  established  to  which  the  claims  are 
tied  in  the  same  manner.  These  monuments  may  be  natural 
objects,  as  mountain  peaks,  permanent  rocks,  or  they  may  be 
artificially  constructed,  in  which  case  they  should  be  strong  and 
without  liability  to  disturbance.  The  method  of  tying  to  such  a 
monument  is  shown  in  Fig.  179.  When  a corner  is  properly 
tied  in  the  manner  mentioned,  the  courses  are  staked  out, 
1500  ft.  along  the  length  of  the  vein  by  300  ft.  on  each  side  of 
the  same.  The  end  lines  must  be  parallel  to  each  other  in 
accordance  with  the  Mining  Law,  but  the  side  lines  need  not  be 
so,  though  the  survey  of  a lode  “must  be  substantially  a 
parallelogram.” 

377.  Before  commencing  the  survey  for  a patent,  the  original 
stakes  are  found,  if  possible ; that  is,  if  no  amended  location  cer- 
tificate has  been  filed.  In  the  latter  case,  the  stakes  marking 
the  amended  location  are  the  ones  to  be  followed.  When  no 
stakes  can  be  found,  the  surveyor  will  be  guided  by  the  locator  or 
some  one  familiar  with  the  boundaries  of  the  property,  provided 
the  boundaries  then  pointed  out  do  not  differ  from  the  record  of 
the  claim  on  file  in  the  Recorder’s  office.  Even  in  the  latter 
case  the  surveyor  can,  at  the  wish  of  the  owner,  and  when  there 
is  no  interference  with  other  locations,  alter  the  claim  (swing  it), 
file  an  amended  location  notice  in  the  Recorder’s  office,  and  send 
forward  the  description  and  maps  of  the  altered  claim  for 
patent. 

378.  In  making  a survey,  it  is  frequently  found  that  there 
arc  conflicting  claims,  that  is,  two  or  more  claims  cross  or  lap 


SEC.  II.] 


METHOD  OF  LOCATING  CLAIMS. 


343 


over  each  other.  An  example  is  shown  in  Fig.  180,  which  is  a 
plot  made  from  actual  claims  on  Treasure  Hill,  Black  Hills, 


Dakota.  When  claims  conflict,  priority  of  location  must  govern 
as  to  the  ownership  of  the  surface  in  dispute,  and  the  U.  S. 
Deputy  Surveyor  in  making  his  plot,  etc.,  is  required,  by  the 
land-office  instructions,  to  show  and  deduct  the  area  in  conflict 
from  the  subsequent  location, — as  is  shown  in  the  figure.  Should 
this  not  be  satisfactory  to  the  owner,  his  appeal  is  to  the 
courts. 


344 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


379.  When  the  survey  for  patent  is  completed,  the  surveyor 
drives  strong  stakes,  generally  about  4 inches  to  5 inches  square, 
and  4 to  5 feet  long,  firmly  into  the  ground,  and  marks  them 
M.  S.  (Mineral  Survey)  No.  1,  M.  S.  No.  2,  etc.  No  less  than 
4 stakes,  one  in  each  corner,  can  be  used  to  mark  a claim,  and  in 
some  cases  more  are  needed.  In  Fig.  180,  for  example,  it  is 
readily  seen  that  four  stakes  would  not  be  sufficient  to  trace  out 
the  boundaries  of  the  Golden  Seal,  Ocean  Wave,  Elkhorn,  or 
Fenian  claims. 

380.  When  claims  are  surveyed  for  patent,  they  are  numbered 
in  the  order  of  their  application,  Lot  No.  1,  No.  2,  No.  3,  etc. 
Figure  180  shows  that  but  3 of  the  claims  represented  were 
patented  at  the  time  the  survey  was  made,  viz.,  Placer  Lot  No. 
120  and  55,  and  Esmeralda  Lot  No.  226.  The  Esmeralda  being 
patented,  it  will  be  seen  that  the  areas  of  the  other  claims  in 
conflict  with  it  have  been  deducted,  so  also  of  the  others  in  the 
order  of  their  priority. 

381.  With  regard  to  errors  and  corrections  in  field  and  office 
work,  in  a survey  for  patent,  the  following  instructions  are  given 
to  Deputies  (see  Special  Instructions  to  Deputies  by  the  Survey- 
ors-General  of  Montana  and  New  Mexico) : 

“ The  error  in  balancing  must  not  exceed  two  feet  in  one 
thousand  feet,  and  this  will  be  allowed  only  on  complicated  sur- 
veys or  when  the  ground  is  very  rough.” 

In  the  notes  of  survey  returned,  must  be  a table  of  area  cal- 
culated by  double  meridian  distances;  in  this  table  must  be 
shown  first  the  balance  of  the  survey  as  actually  run  upon  the 
ground,  and  immediately  thereafter  the  corrected  latitudes  and 
departures.  The  correction  must  bo  made,  either  by  distributing 
the  error  among  the  courses,  in  proportion  to  the  length  of  each 
course,  by  the  proportion — The  sum  of  the  lengths  of  all  the 
courses  is  to  the  total  error  in  latitude  or  departure  as  the  length 


SEC.  II.] 


METHOD  OF  LOCATING  CLAIMS. 


345 


of  each  course  is  to  the  correction  of  its  latitude  or  departure 
(see  Art.  127) ; or  by  the  following  method,  which  is  considered 
preferable : 

“You  can  correct  the  balance  by  computing  a closing  course 
and  distance.  This  can  be  made  on  the  last  line  of  the  survey 
when  it  is  one  of  the  longest  lines,  but  otherwise  should  be  made 
upon  the  longest  line  unless  there  should  be  a line  that  could  not 
be  run  accurately,  and  in  that  case  close  on  it,  when  you  use  this 
method.  This  corrected  line  will  be  the  true  line  to  be  used  in 
the  notes  and  on  plot.  In  notes  you  will  state  immediately  fol- 
lowing (parenthetically)  the  course  and  distance  of  the  line  as 
actually  run  upon  the  ground.  This  is  the  better  way,  as  the 
actual  error  made  in  the  survey  can  be  detected  by  reference  to 
this  line  only.” 

“ The  maximum  error  that  will  be  allowed  in  a lode  claim  is 
0.03  acres.” 

382.  The  instructions  of  the  U.  S.  General  Land  Office  to 
U.  S.  Deputy  Mineral  Surveyors  are  embraced  in  a series  of 
decisions  and  letters  to  the  U.  S.  Surveyors-General,  relating  to 
particular  cases.  Based  upon  these  decisions  and  letters,  and 
upon  what  in  their  judgment  and  experience  seemed  necessary, 
the  Surveyors-General  have  generally  issued  printed  instructions 
to  their  deputies. 

A copy  of  the  instructions  of  the  Surveyor-General  of 
Colorado  to  Deputy  Mineral  Surveyors,  with  sample  Field 
Notes  and  sample  Plot  furnished  therewith,  is  given  in  Ap- 
pendix C. 


346 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


SECTION  III. 

UNDERGROUND  TRAVERSING,  ETC. 

383.  Let  it  be  required  to  sink  a shaft  from  the  surface  to 
connect  with  a tunnel,  which  is  driven  into  a hillside. 

In  mauy  cases  the  needle  cannot  be  used  on  account  of 
disturbing  influences,  such  as  the  proximity  of  iron  and  its  ores, 
or  magnetic  pyrites.  The  vernier  readings  of  the  transit  are  then 
employed,  and  indeed  this  method  is  preferable  in  all  cases. 

Ascertain  the  course  of  the  tunnel  and  the  position  of  the 
point  where  it  is  desired  to  make  the  connection.  Establish  a 
station-stake  at  any  convenient  distance  from  the  mouth  of  the 
tunnel  that  will  command  a sight  of  its  centre  point.  If  the 
centre  line  of  the  tunnel  is  a straight  line,  or  nearly  so,  a sight 
may  be  taken  from  the  established  station  to  the  point  of  inter- 
section. In  general,  however,  this  cannot  be  done,  the  line  of 
sight  being  cut  off  by  deflections  in  the  course,  thus  requiring  a 
traverse. 

384.  Where  curves  occur,  stations  should  be  established  by 
driving  nails  in  the  centre  of  the  cap-timber,  overhead  in  the 
tunnel.  If  there  are  no  timbers,  holes  should  be  drilled  in  the 
rock  and  filled  with  wooden  spuds  into  which  nails  are  driven. 
Plumb-lines  are  then  suspended  from  these  nails,  lighted  by 
lamps  suitably  placed,  and  used  for  sighting  to  with  the 
transit. 

If  desired,  the  selected  stations  may  be  marked  by  iron  pegs 
driven  into  holes  drilled  in  the  floor  of  the  tunnel.  In  this  case, 
guiding  lamps  placed  on  stands  similar  to  tripods,  but  with 
sliding  pieces  carrying  the  lamps  and  fixed  in  position  by  clamp- 
screws,  are  used  for  sighting  to.  By  the  sliding  and  clamp 


3EC.  III.] 


UNDERGROUND  TRAVERSING. 


347 


arrangement,  the  height  of  the  lamp  may  be  made  equal  to  that 
of  the  line  of  sight  of  the  transit  telescope. 

As  the  same  methods  will  apply  in  either  case,  the  suspended 
plumb-lines  will  be  assumed  as  giving,  in  general,  better 
results. 

385.  To  place  the  transit  accurately  in  position  at  a station, 
take  a block  of  lead  with  a steel  point  in  the  centre  projecting  out 
about  half  an  inch ; place  it  on  the  floor  of  the  tunnel,  with  the 
steel  point  upward,  immediately  under  a plumb-bob  let  down 
from  the  station  overhead ; the  plumb-line  may  then  be  removed 
and  the  transit  set  in  position  over  the  steel  point. 

386.  Figure  181  shows  a somewhat  extreme  case,  chosen  to 
illustrate  the  method  of  traversing.  Take  a station  at  A , out- 
side and  100  feet  (say)  from  the  mouth  of  the  tunnel.  Fix  a 
stake  at  A,  and  drive  a nail  in  the  top  of  it  for  convenience  and 
accuracy  of  sighting.  Calling  the  outside  station  No.  1,  and  the 
one  at  the  mouth  of  the  tunnel  No.  2,  take  the  line  between 
stations  1 and  2 as  the  meridian  of  the  survey,  find  the  azimuths 
of  the  several  successive  courses,  2 to  3,  3 to  4,  &c.,  with  respect 
to  this  meridian,  precisely  as  directed  in  Art.  194.  The  bearing 
of  the  several  courses  with  respect  to  this  assumed  meridian  may 
then  be  found  and  tabulated  as  directed  in  Art.  195. 

If  necessary,  measurements  are  made  to  determine  the  cross- 
section  of  the  tunnel,  its  height  or  breadth,  at  a station,  or  at  any 
desired  point  of  the  course,  and  the  results  entered  in  the  column 
of  remarks  in  the  field-book. 

387.  The  distances  between  the  stations  are  now  measured 
with  great  care.  A chain  should  not  be  used  in  these  measure- 
ments ; a light  steel  tape  is  much  better.  The  points  corres- 
ponding to  the  ends  of  the  tape  are  marked  by  chalk  lines  on  the 
rock,  or  in  some  other  convenient  manner.  The  distances  may 


348 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


be  measured  horizontally,  or  along  the  slope  Of  the  tunnel,  in 
which  latter  case  they  must  be  reduced  to  horizontal  distances. 


388.  When  distances  between  stations  are  measured  along  the 
slope,  the  angle  of  elevation  or  depression  of  the  course  must  be 
determined — and  may  be  determined  as  follows: 


Fig.  181. 


SEC.  III.] 


UNDERGROUND  TRAVERSING. 


349 


Let  the  nail  on  staff  at  A (Fig.  181)  be  at  the  same  distance 
above  the  ground  at  A that  the  axis  of  the  vertical  limb  of  the 
transit  is  above  the  floor  of  the  tunnel  at  station  2 ; when  the 
telescope  is  pointed  to  the  nail  at  A,  the  reading  of  the  vertical 
limb  will  be  the  angle  of  inclination  of  the  first  course,  in  eleva- 
tion, if  the  telescope  points  downward — in  depression , if  it  points 
upward.  Let  the  point  of  a plumb-bob,  suspended  from  the 
nail  overhead  at  station  3,  at  the  same  distance  above  the  floor  of 
the  tunnel  at  3 that  the  axis  of  the  vertical  limb  of  the  transit  is 
above  the  floor  at  2 ; when  the  telescope  is  directed  to  the  point 
of  the  plumb-bob,  the  reading  of  the  vertical  limb  will  be  the 
angle  of  inclination  of  the  second  course,  in  elevation,  if  the 
telescope  points  upward — in  depression,  if  it  points  downward. 
In  like  manner  are  obtained  the  angles  of  elevation  or  depression 
of  the  several  successive  courses  to  the  end  of  the  traverse. 

389.  The  method  of  recording  the  observations  and  measure- 
ments made  in  a traverse,  is  shown  in  the  following 


Field  Book. 


Stations. 

Angles  op  Inclination. 

Azimuth  with 

Distance 
in  feet. 

Remarks. 

Elevation. 

Depression. 

Course  1.2. 

1 

O 

o 

© 

O 

o 

© 

© 

o 

© 

102 

Sta.  1,  at  staff,  outside 

tunnel. 

2 

© 

o 

© 

100 

Sta.  2,  at  iron  peg,  at 

3 

340° 

150 

mouth  of  tunnel 

overhead. 

4 

300° 

180 

5 

280° 

200 

At  Sta.  5,  breadth  GJ  ft. 

6 

325° 

120 

7 

275° 

162 

At  Sta.  7,  height  7*  ft. 

8 

265° 

120 

At  iron  peg. 

9 

350 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


The  above  is  a record  of  the  traverse  shown  in  Fig.  181. 
The  distances  measured  are  supposed  to  be  horizontal  distances, 
and  the  angles  of  inclination  are,  therefore,  zero. 

390.  The  method  of  reducing  the  traverse  is  as  follows . 

From  the  azimuths  of  the  several  courses  with  the  given 
course  taken  as  meridian,  find  the  bearings  (see  Art.  195) 
and  proceed  as  shown  in  the  following 

Office  Form. 


Station.  j 

Slope  in  Feet. 

Bearing  with 
Course  1.2. 

Length 

or 

Course. 

Latitude. 

Depaktuue. 

Eleva- 

tion. 

Depres- 

sion. 

N. 

s. 

E. 

W. 

1 

0 

0 

North 

102 

102 

2 

North 

100 

100 

3 

N 20°  W 

150 

140.95 

51.30 

4 

N 60°  W 

180 

90 

155.88 

5 

£ 

0 

O 

QO 

£ 

200 

34.72 

196.96 

6 

N 35°  W 

120 

98.30 

68.83 

7 

N 85°  W 

162 

14.12 

161.38 

8 

S 85°  W 

120 

10.46 

119.54 

9 

a 

580.09 

10.46 

00 

753.89 

■2 

10.46 

00 

V 

K 

N 52°5G'W 

944.68 

569.63 

753.89 

Hence  the  resultant  course  AC,  from  1 to  9,  has  a northing 
of  5G9.63  feet  and  a westing  of  753.89;  to  find  its  length  and 
bearing,  we  have  in  the  triangle  ABC, 

AB  = 753.89  and  BC  = 5G9.63  ; 
hence  AC  = ^ AB*  + DC2  = 944.89. 


SEC.  III.] 


UNDERGROUND  TRAVERSING. 


351 


n BC  569.63 
tan  ^(7 


BAC=  37°  4'; 

. bearing  of  AO  = 90°  — 37°  4'  = 52°  56'. 

391.  If  the  distances  between  stations  are  measured  on  the 
slope,  instead  of  horizontally,  the  angle  of  elevation  or  depres- 
sion of  each  course  must  be  observed,  and  recorded  in  the  proper 
column  of  the  field  book.  The  distances  on  the  slope  must 
then  be  reduced  to  horizontal  distances  in  finding  the  resultant 
course.  The  method  of  record  and  reduction  in  such  a case, 
and  when  the  magnetic  bearing  of  the  course,  from  which 
azimuths  are  measured,  is  given,  will  be  apparent  from  the 
following 

Example.  Required  the  resultant  of  the  traverse  of  which 
the  notes  are  contained  in  the  following 


Field  Book. 


Stations. 

Angles  of  Inclination. 

Azimuth  with 
Course  1.2. 

Distance 
in  feet. 

Remarks. 

Elevation. 

Depression. 

1 

2°  30' 

© 

o 

O 

307 

Sta.  1,  at  iron  peg, 

centre  of  shaft. 

2 

3°  15' 

176°  15' 

402 

3 

3°  30' 

© 

CO 

o 

00 

o* 

e* 

240 

4 

4°  15' 

© 

G 

367 

5 

1°  30' 

246°  15' 

409 

6 

2°  00' 

249°  45’ 

200 

At  iron  peg. 

7 

Let  the  magnetic  bearing  of  the  course,  1.2,  from  which 
azimuths  are  measured,  be  S 23°  W. 


ELEMENTS  OF  SURVEYING. 


[HOOK  X. 


Find,  from  the  bearing  of  the  course  taken  as  meridian  and 
the  azimuths  of  the  several  courses,  the  bearings  of  the  several 
courses,  as  in  Art.  196.  The  traverse  is  then  reduced  as  in  the 
following 

Office  Form. 


Course. 

Slope  in  Feet. 

Bearing. 

Reduced 

length 

of 

course. 

Latitude. 

Departure. 

Eleva- 

tion. 

Depres- 

sion. 

N. 

s. 

E. 

W 

1 

13.4 

S 23°  W 

306.7 

282.3 

119.8 

2 

22.8 

S 19J°W 

401.4 

378.9 

132.3 

3 

14.7 

S 7H°  W 

239.6 

76.0 

227.2 

4 

27.2 

N40°  W 

366. 

280.4 

235.3 

5 

10.7 

S 89i°  W 

408.9 

5.4 

408.9 

6 

7. 

N87i°W 

199.9 

9.6 

199.7 

17.7 

78.1 

290. 

742.6 

1323.2 

17.7 

290. 

0.0 

Result- 

ant. 

60.4 

S71°7'W 

1406.9 

452.6 

1323.2 

The  length  of  the  course  on  the  slope,  multiplied  by  the  sine 
of  the  angle  of  inclination,  gives  the  distance  the  course  rises  or 
falls,  in  feet  (if  the  measurements  are  made  in  feet)  ; and  the 
length  multiplied  by  the  cosine  of  the  angle  of  inclination,  gives 
the  reduced  length  of  the  course,  that  is,  the  length  that  would 
have  been  found  had  the  course  been  measured  on  a horizontal 
line.  The  bearing  and  reduced  length  of  each  course  being 
found,  the  latitudes  and  departures  of  the  several  courses  are 
found  as  before.  In  this  example,  the  resultant  course  descends 
00.4  feet,  its  southing  is  452.6  feet,  and  its  westing  1323.2;  its 
bearing,  S.  71°  7'  W.,  and  length  on  the  horizontal,  1406.98,  are 
then  found  as  before. 

392.  The  difference  of  level  of  two  points  underground  may 


SEC.  III.] 


UNDERGROUND  TRAVERSING. 


353 


be  found  as  shown  in  the  example  given  in  Art.  391,  or  by 
running  a line  of  levels  between  the  points  as  directed  in 
Art.  294.  Leveling  rods  for  underground  use  should  be 
painted  and  graduated  to  feet,  tenths,  and  hundredths  of  a 
foot,  and  about  3£  to  5 feet  in  length.  When  reading  the 
graduations  of  the  rod  through  a telescope,  the  flame  of  two 
lamps,  one  from  each  side,  should  be  brought  to  bear  on  its  face 
so  as  to  light  it  and  not  obstruct  the  line  of  sight  of  the  observer. 


Method  of  Plotting  the  Underground  Traverse  on  the  Surface. 

393.  To  plot  the  traverse  on  the  surface  of  the  earth,  the 
first  course,  from  which  azimuths  were  measured,  must  first  be 
laid  down,  in  direction  and  horizontal  length,  and  its  two  ends 
marked  with  suitable  pegs.  If  the  first  station  has  been  taken 
outside  the  tunnel,  as  in  the  traverse  shown  in  Fig.  181,  the  first 
course  is  already,  or  may  easily  be,  pegged  out  on  the  surface. 
If,  however,  the  first  course  is  within  the  tunnel,  as  in  example 
given  in  Art.  391,  the  first  course  must  be  connected  with  the 
surface.  There  are  two  principal  methods  of  making  the  con' 
nection : 

394. FIRST  METHOD. 


A straight-edge,  A B, 
is  mounted  on  two 
trestles,  and  from  it  are 
suspended  two  plumb 
lines,  E and  F,  as  far 
apart  as  the  breadth  of 
the  shaft  will  permit. 
To  prevent  agitation 
from  currents  of  air,  the 
bobs  are  permitted  to  dip 
into  buckets  of  water,  at 


Fig.  182. 


354 


ELEMENTS  OF  SURYEYINO. 


[BOOK  X. 


the  bottom  of  the  shaft ; the  transit  being  at  the  second  station, 
K,  and  the  telescope  turned  in  the  direction  of  the  first  station, 
D , the  straight-edge  is  moved  by  an  assistant  until  both  are  seen 
in  line  from  K ; their  plane  then  passes  through  the  first  course; 
and  if  the  line  AB  be  prolonged  to  if  and  L , the  line  ML  will  be 
directly  over  the  first  course,  and  consequently  its  bearing  will  be 
that  of  the  first  course.  By  measuring  the  line  E,  the  depth  of 
the  shaft  may  be  found. 


395. SECOND  METHOD. 


Let  the  transit,  provided 
with  a diagonal  eye-piece, 
be  planted  over  the  station 
D,  at  the  foot  of  the  shaft, 
and  after  being  leveled,  let 
it  be  directed  on  the  sta- 
tion K.  Then,  without 
changing  the  plane  of 
vision,  let  the  transit  be 
directed  to  the  top  of  the 
shaft,  and  let  an  assistant  plant  two  flag  rods,  one  at  A and  the 
other  at  B , both  in  the  plane  of  vision,  and  let  the  line  AB  be 
prolonged  to  L and  M,  as  before.  The  line  LM  will  be  in  the 
same  vertical  plane  with  the  first  course,  DK.  Hence,  as  before, 
we  may  determine  the  bearing  of  the  first  course  of  the  traverse. 

396.  Having,  on  the  surface,  the  direction  of  the  first  course 
and  the  position  of  Station  1 of  the  traverse,  Station  2 of  the 
traverse  may  be  found  by  laying  off  from  Station  1,  in  the  proper 
direction,  a horizontal  distance  equal  to  the  length  of  the  first 
course.  The  position  of  Station  2 on  the  surface  will  thus  be 
determined.  Put  the  transit  over  the  surface  position  of  Station 
2,  make  the  zero  of  the  vernier  coincide  with  the  zero  of  the 


SEC.  III.] 


UNDERGROUND  TRAVERSING. 


355 


horizontal  limb  and  clamp  the  vernier  plate ; direct  the  telescope 
to  surface  Station  1 and  clamp  the  limb  ; revolve  the  telescope  on 
its  horizontal  axis,  unclamp  the  vernier  plate  and  revolve  it 
around  to  the  right,  through  the  azimuth  of  the  course,  from  2 
to  3 ; the  line  of  sight  of  the  telescope  will  then  be  in  the  direc- 
tion of  the  traverse  line  from  Station  2 to  Station  3 ; measure 
from  surface  Station  2,  in  the  direction  found,  a horizontal 
distance  equal  to  the  horizontal  distance  of  the  traverse  lino 
from  Station  2 to  Station  3 ; the  extremity  will  be  the  surface 
position  of  Station  3.  Clamp  the  vernier  plate  and  remove  the 
transit  to  surface  Station  3;  reverse  the  telescope  on  its  hori- 
zontal axis,  loosen  the  lower  clamp  and  sight  surface  Station  2; 
tighten  the  lower  clamp  and  revolve  the  telescope  on  its  axis ; 
unclamp  the  vernier  plate  and  turn  it  through  the  azimuth  of 
the  course  from  3 to  4 ; the  line  of  sight  of  the  telescope  will 
then  lie  in  the  direction  of  the  traverse  line  from  Station  3 to 
Station  4 ; determine  the  surface  position  of  Station  4 by 
measurement  from  surface  Station  3,  as  before  (see  Art.  196). 
Proceed,  in  like  manner,  to  mark  out  on  the  surface  the  suc- 
cessive underground  stations  and  courses,  till  the  required  point 
(9  in  Fig.  181)  is  found. 

397.  The  slope  rod  is  a convenient  instrument  for  use  in 
making  surface  measurements. 

A device  for  such  a rod,  for  which  caveat  was  entered  by  A.  J. 
Rigby  of  New  York  in  1883,  consists  essentially,  as  shown  in 
Fig.  184,  of  a horizontal  bar  10  to  12  feet  long,  made  to  move  up 
and  down  a vertical  bar  6 to  8 feet  long,  to  which  it  may  be 
clamped  in  any  position. 

The  horizontal  bar,  A,  carries  a scale,  graduated  to  hun- 
dredths of  a foot  and  reading  by  vernier  to  thousandths  of  a foot, 
a spirit  level  E , by  which  the  bar  may  be  made  truly  horizontal, 
and  an  extension  bar  6r,  which  may  bo  drawn  out  4 to  6 feet, 


356 


ELEMENTS  OF  SURVEYING. 


[BOOK  X. 


making  the  extended  bar  14  to  18  feet  long  ; when  the  bar  is  drawn 
out,  the  reading  is  made  by  a vernier  on  the  extension  bar,  as 
shown  in  the  figure,  in  a manner  similar  to  that  in  which  the 


extended  leveling  rod  is  read  (see  Art.  290).  The  vertical  bar, 
B , also  carries  a graduated  scale,  reading  by  vernier  to  thou- 
sandths of  a foot.  The  two  clamp-screws,  FF,  working  against 
a clamp-plate,  fasten  the  horizontal  to  the  vertical  bar  and  keep 
it  firmly  in  position,  both  horizontally  and  vertically,  against  its 
bearings  on  the  vertical  bar. 

The  manner  of  using  the  slope-rod  is  shown  in  Fig.  185. 
Pegs  are  driven  into  the  ground  at  convenient  stations  along  the 
line  which  is  to  be  measured,  the  exact  point  on  each  being 
marked  by  the  intersection  of  cross-lines  on  the  top  of  the  peg ; 
if  the  selected  station  is  on  a rock,  and  a peg  can  not  be  driven, 
the  cross-lines  are  marked  in  the  rock  to  note  the  station.  The 
front  edge  of  the  vertical  bar  is  placed  at  the  point  marked  on 
one  peg  and  the  end  of  the  horizontal  bar,  extended  if  necessary, 
to  the  point  marked  on  the  other ; the  horizontal  bar  is  moved 


sec.  hi.] 


UNDERGROUND  TRAVERSING. 


357 


slowly  up  and  down  the  vertical  bar  till  the  bubble  of  the  spirit- 
level  E is  in  the  middle  of  its  run,  and  then  clamped  in  position. 


The  reading  on  the  horizontal  bar  will  give  the  true  horizontal 
distance  between  the  two  stations,  and  the  reading  on  the 
vertical  bar  will  give  their  difference  of  level. 

398.  A better  method  than  repeating  the  traverse  on  the 
surface  will  appear  from  a consideration  of  the  diagram,  Fig.  181. 
It  will  be  seen  that  the  algebraic  sums  of  the  latitudes  and 
departures  respectively  of  the  several  courses,  with  respect  to 
the  first  course  taken  as  a meridian,  are  equal  to  the  latitude  and 
departure  of  the  last  or  station  9.  The  latitude  of  9 is  the  perpen- 
dicular of  a right-angled  triangle  and  the  departure  is  the 
base ; the  distance  A C,  from  1 to  9,  is  equal  to  the  square  root 
of  the  sum  of  the  squares  of  the  latitude  and  departure  of  9. 
The  angle  BAC  may  be  found  by  the  relation 

tan  A = (Art.  35). 

Having  found  the  angle  BAC,  the  bearing  of  AC,  with 
respect  to  the  course  1.2  taken  as  meridian,  is  equal  to 
$0°  — BA  C.  Place  the  transit  over  station  1,  level  it,  make  the 
zero  of  the  vernier  coincide  with  the  zero  of  the  horizontal  limb. 


358  ELEMENTS  OF  SURVEYING.  [ROOK  X. 


263  jL 1 3*o_& 

VERTICAL  SECTION  THROUGH  A.B.O. 


SEC.  III.] 


UNDERGROUND  TRAVERSING. 


359 


clamp  the  vernier  plate,  sight  to  station  2 and  clamp  the  limb ; 
unclamp  the  vernier  plate,  turn  it  through  the  bearing,  as  found, 
of  A C,  and  clamp  it ; the  line  of  sight  of  the  telescope  will  be  in 
the  line  AC.  Measure  the  horizontal  distance,  as  found,  from  A 
to  C\  and  the  point  9 will  be  accurately  determined. 

Again,  from  1 measure  off  the  departure  of  9 at  right  angles 
to  the  assumed  meridian,  from  the  end  of  which,  and  at  right 
angles  to  it,  measure  off  the  latitude ; the  end  of  this  latter  line 
will  be  the  required  point  9. 

The  surveyor  should  be  satisfied  with  nothing  less  than  per- 
fect coincidence  of  the  points  so  determined.  Fig.  186  shows 
sectional  maps  of  work  performed  in  this  manner. 


399.  To  produce  a line,  previously  marked  out  on  the  sur- 
face between  two  shafts,  in  the  same  relative  direction  below 
ground,  so  as  to  form  a heading  or  tunnel  from  one  shaft  to 
another,  proceed  in  the  following  manner : 

Set  up  the  transit  at  the  bottom  and  in  the  centre  of  the 
shaft,  as  near  as  can  be  estimated  by  the  eye,  as  at  a , Fig.  187, 
and,  after  the  instrument  has 
been  accurately  leveled  and 
the  zero  of  the  vernier  made 
to  coincide  with  the  zero  of 
the  horizontal  limb,  sight 
up  the  shaft  and  make  the 
cross- wires  of  the  telescope 
bisect  a mark  b,  in  the  line 
A B at  the  surface  ; revolve 
the  telescope  vertically  a little  until  the  vertical  wire  strikes  the 
opposite  side  of  the  shaft  at  c.  Measure  the  angle  cbe , or  its  equal 

<(?- , and  the  radius  of  the  shaft  ae.  The  exact  deviation  of  the 


centre  of  the  transit  from  the  vertical  plane  of  the  line  AB  at 
the  surface,  may  then  be  found  by  calculation  thus:  Let  the 


360 


ELEMENTS  OF  SURVEYING. 


BOOK  X. 


angular  deviation,  or  angle  cae , be  3°  10',  and  the  radius  of  the 
shaft  60  inches,  then 

sin  — x ac  = .27639  x 60  = 16.5, 

/i 

the  deviation  required  in  inches.  Remove  the  transit  from  a 
along  the  line  ss',  and  toward  a distance  of  16.5  inches ; it 
will  then  occupy  its  true  position,  or  be  in  the  same  vertical 
plane  as  the  line  AB  at  the  surface. 

The  transit  may  now  be  adjusted  and  leveled,  and  the  tele- 
scope pointed  up  the  shaft ; when,  if  the  preceding  operation  has 
been  properly  performed,  the  vertical  wire  will  exactly  pass 
through  the  marks  previously  fixed  at  e and  b (H.  D.  Hoskold). 
Revolving  the  telescope  down,  its  line  of  sight  will  give  the  true 
direction  below  ground  of  the  line  AB  at  surface. 

Method  of  Plotting  the  Traverse  on  Paper. 

400.  To  plot  the  traverse  on  paper,  we  first  plot  the  plan 
by  the  usual  method  of  plotting  compass-work,  using  the  bear- 
ings and  the  reduced  lengths  of  the  courses.  This  gives  the 
general  direction  of  the  horizontal  projection  of  the  traverse 
run  ; and  from  the  measurements  for  cross-section,  the  breadths 
of  the  tunnel  on  each  side  may  be  plotted,  ancf  thence  a 
complete  plan  of  the  mine  may  be  constructed.  We  next  plot 
the  profile  of  the  traverse,  using,  as  in  railroad  plotting,  two 
scales,  one  for  horizontal  distances,  and  the  other  and  larger 
one  for  vertical  distances.  The  relation  between  the  two  scales 
will  depend  upon  the  circumstances  of  the  case.  Sometimes, 
both  may  be  equal.  The  profile  represents  the  undulation  of 
the  traverse,  without  reference  to  its  horizontal  deviations.  Let 
us  conceive  vertical  planes  to  be  passed  through  all  the  courses. 
These  will  intersect  each  other  in  vertical  lines.  Take  the  one 
through  the  first  course,  as  the  one  on  which  the  profile  is  to 


SEC.  IV.] 


PRACTICAL  APPLICATIONS. 


361 


be  delineated.  Then,  beginning  with  the  plane  through  the 
last  course,  conceive  the  other  planes  to  be  revolved,  in  order, 
each  about  its  intersection  with  the  preceding  one,  to  coincide 
with  it,  and  so  on  till  all  are  brought  into  coincidence  with 
the  fixed  one.  The  lines  of  the  traverse  will  then  be  situated 
in  one  plane,  and  a plot  of  them,  in  this  position,  will  be  the 
profile  required.  The  distances  from  the  traverse  to  the  floor 
and  roof  of  the  tunnel,  at  different  points,  enable  us  to  com- 
plete the  profile. 


SECTION  IV. 

PRACTICAL  APPLICATIONS. 

401.  To  determine  the  position  and  depth  of  a mine  shaft, 
CB , Fig.  188,  which  is  to  connect  with  a tunnel  AB. 


Fig.  188. 


If  the  course  of  the  tunnel  is  not  a straight  line  it  must  be 
traversed.  Assuming  it  to  be  a straight  line,  the  bearing  is 
obtained  with  the  transit.  The  length  of  the  tunnel  is  then 
accurately  measured  with  a steel  tape  or  a rod.  The  surveyor 
now  stakes  out  the  course  on  the  surface,  then  beginning  at  the 


3G2 


elements  of  surveying. 


[book  x. 


mouth  of  the  tunnel  the  first  peg  is  driven  on  a level  with  the 
door,  and  the  measurement  of  the  tunnel,  say  1000  feet,  is  laid 
off  on  the  course  established  on  the  surface.  If  the  slope  rod  is 
used,  the  sum  of  the  vertical  measurements  or  readings  will  be 
the  vertical  distance  of  C above  A.  Should  the  tunnel  at  B be 
higher  or  lower  than  at  A,  the  difference  subtracted  from  or 
added  to  the  sum  of  the  vertical  readings,  will  be  the  required 
depth  of  the  shaft.  If  the  slope-rod  is  not  used,  a line  of  levels 
must  be  run  to  find  the  difference  of  level  between  A and  C. 


402.  To  find  the  distance  necessary  to  connect  the  main 
shaft  GF  and  the  tunnel  EF  (Fig.  189). 


Fig.  189. 


Determine  the  depth  of  the  shaft  DE,  by  suspending  a chain 
from  the  surface  or  by  a steel  tape  with  a weight  attached. 
Measure  the  length  of  the  tunnel  EH,  already  completed,  and, 
with  the  slope  rod,  or  otherwise,  obtain  the  horizontal  distance 
DJ,  and  the  difference  of  level  between  the  points  D and  G. 
This  difference  of  level  added  to  the  depth  of  the  shaft  DE  = the 
total  depth  of  the  shaft  GF.  Find  the  depth  of  the  shaft  GJ 


the  required  distances.  In  this  case  the  tunnel  is  assumed  to  be 


SEC.  IV.] 


PRACTICAL  APPLICATIONS. 


363 


straight  and  the  position  of  the  shaft  determined  as  in  the 
preceding  problem. 


403.  To  find  the  depth  of  a mine  shaft  AB,  Fig.  190,  the 
distance  from  the  outcrop  at  C and  the  dip  of  the  vein  CB 
being  given. 

Determine  the  difference 
of  level  AD  = 100  ft. 

The  horizontal  distance 
CD  = 600  ft. 

Angle  DCB  = 60°. 

Then  DB  = CD  tan  60° 

= 1039.2  ft. 

And  AB  — AD  + DB 
= 100  + 1039.2  = 1139.2. 

To  find  the  distance 
from  the  outcrop  to  the 
point  of  intersection,  or  the 
distance  CB,  we  have  (Da- 
vies’ Leg.,  Trig.,  Art.  37), 


CB  = 


CD 

cos  60° 


5 = ^=1200. 


Fig.  190. 


by  construction. 

Draw  to  any  suitable  scale  the  distance  CD  = 600  feet ; from 
C draw  the  line  CD,  making  an  angle  of  60°  with  the  line  CD  ; 
from  the  point  D draw  a line  perpendicular  to  CD,  until  it  inter- 
sects CB  at  B;  produce  it  upward  to  A,  until  distance  from  D 
to  A = difference  of  elevation  between  C and  A.  Measure  the 
line  AB  on  the  scale,  which  gives  the  required  depth.  Measure 
also  CB  on  the  scale,  which  is  the  distance  from  the  outcrop  to 
the  point  of  intersection. 

404.  Given  the  depth  of  the  shaft  DE  = 588.5  feet,  Fig. 
191,  and  the  horizontal  distance  FD  from  the  outcrop  to  the  top 


ELEMENTS  OF  SURVEYING. 


364 


[book  X. 


of  shaft  = 500  feet,  and  the  angle  GFD  = 60°,  to  find  the 
length  of  a cross-cut  EG 
from  the  bottom  of  the 
shaft  to  intersect  the  vein  ; 
also  the  distance  from  the 
outcrop  to  the  point  of 
intersection. 

Find  the  horizontal 
distance  from  the  outcrop 
F,  at  which  a shaft  of 
588.5  feet  will  intersect 
the  vein,  or  the  distance 
Fff.  This  is  readily  found; 
for  assuming  it  to  be  the 
base  of  a right-angled  tri- 
angle, we  have  (Davies’  Leg.,  Trig.,  Art.  37), 


Fig.  191. 


FH  = = 339.78. 

tan  60 


If  a depression  or  elevation  from  the  outcrop  occurs  at  the  point, 
it  must  be  subtracted  from  or  added  to  the  depth  of  the  shaft ; 
for  example,  if  there  is  a depression  of  50  ft.  it  will  only  be  neces- 
sary to  sink  538.5  ft.  to  the  point  of  intersection.  The  horizontal 
distance  so  found,  subtracted  from  the  real  distance  of  the  shaft 
from  the  outcrop  = length  of  cross-cut  = 160.22,  and 

FG  = VDE*  + (FD  - GEf  = 679.5. 


BY  CONSTRUCTION. 

Draw  the  line  FD  = 500  feet  to  any  convenient  scale.  From 
F draw  the  line  FG , making  the  angle  DFG  = 60°.  Draw 
DE  = 588.5  ft.,  and  from  E draw  EG  parallel  to  FD.  The  dis- 
tances can  now  be  taken  from  the  scale. 


SEC.  IV.] 


PRACTICAL  APPLICATIONS. 


365 


405.  Given  the  angle  LJK  = 60°.  The  distance  JL— 700  ft., 
and  the  difference  of  level  between  J and  H = 100  ft.  Required 
the  depth  of  a shaft  from  H that  will  intersect  a cross-cut  of  350  ft. 
from  the  point  K,  Fig.  192. 


From  the  point  K draw  to  JL  a line  parallel  to  HI ; it  will 
intersect  JL  700 — 350  = 350  ft.  from  the  point  J.  This  line  is 
the  perpendicular  of  a right-angled  triangle,  and  we  have 
it  = 350  tan  60°.  Add  to  this  the  difference  of  elevation, 
which  is  the  required  depth. 

BY  CONSTRUCTION. 

Draw  the  line  JL =700  ft.,  and  from  J draw  JK,  making  the 
angle  LJK  = 60°.  From  K set  off  the  distance  KI , parallel  to 
JL,  and  make  it  350  ft.  From  I raise  the  perpendicular  IL  and 
extend  it  to  H,  making  the  distance  LH  = 100  ft.  of  the  scale. 
The  distance  III  is  now  taken  from  the  scale. 

Problems  of  this  kind  occur  only  where  there  are  inclined 
shafts. 


ELEMENTS  OF  SURVEYING. 


[ROOK  X. 


36G 


406.  Given  the  distance  OL  = 400  ft.,  and  the  angle  LON— 
G0°  ; to  find  the  depth  of  a shaft  at  the  point  of  intersection 
with  the  vein,  and  to  find  the  length 
of  a cross-cut  that  will  again  inter- 
sect the  vein  when  the  shaft  is 
continued  250  ft.  below  the  point  of 
its  intersection,  Fig.  193. 

Draw  the  horizontal  line  OL 
= 400  ft.  Also  ON,  making  the 
angle  LON  = 60°.  Then  400 
tan  60°  = depth  of  shaft  at  the 
point  of  intersection.  Construct 
the  shaft  and  extend  it  to  the  point 
M , 250  ft.  below  the  point  of  inter- 
section, and  draw  the  cross-cut 
MN.  Then  by  similar  triangles, 

OL 


LP 


mn=°Lx 


MN  : PM\ 
PM 


LP 


BY  CONSTRUCTION. 

Draw  OL  = 400  ft.;  also  ON,  making  the  angle  LON=§ 0°  ; 
and  LM  to  the  point  M,  250  ft.  below  the  point  of  intersection  P. 
From  the  scale  take  off  LP  and  MN  the  required  distances. 

In  the  preceding  problems  the  angle  of  60°  was  employed  for 
convenience  of  construction,  but  the  principles  would  be  true  for 
any  other  angle. 

407.  Figures  194,  195,  196,  represent  plan,  longitudinal 
and  transverse  sections  of  a developed  mine. 

The  development  consists  of  a vertical  shaft,  an  inclined 
shaft  following  the  dip  of  the  vein,  and  G one-hundred-foot 
levels. 


Stope 


SEC.  IV.] 


PRACTICAL  APPLICATIONS. 


367 


Line  of  Croppings. 


368 


ELEMENTS  OF  SURVEYING. 


[ROOK  X. 


In  opening  mines  it  is  considered  good  practice  to  follow  the 
vein  for  a considerable  distance  in  depth,  to  be  fully  satislied  of 
its  continuity,  before  sinking  a vertical  shaft  for  deep  working. 

Should  the  vein  be  vertical,  however,  the  prospecting  shaft 
may  be  made  the  working  shaft.  The  depth  to  which  a shaft  is 
sunk  on  the  dip  of  the  vein  will  depend  upon  the  Engineer  in 


Fro.  195. 


charge  and  upon  the  characteristics  of  the  vein.  A careful 
engineer  will  not  incur  the  expense  of  sinking  a deep  working  shaft 
to  intersect  an  inclined  vein,  until  he  has  followed  the  vein  for 
such  a distance  that  the  possibility  of  its  terminating,  or  “pinch- 
ing out,”  is  quite  remote.  Should  the  vein  be  irregular  and  not 
possess  well-defined  fissure  characteristics,  the  greater  the  neces- 
sity for  care  in  this  respect  and,  in  cases  of  great  uncertainty,  it 
would  even  be  advisable  to  have  the  working  shaft  follow  the  dip 


SEC.  IV.] 


PRACTICAL  APPLICATION'S. 


369 


Fig.  196. 


370 


ELEMENTS  OP  SURVEYING. 


[BOOK  X. 


of  the  vein.  If,  on  the  other  hand,  the  vein  is  found  to  he  strong 
and  regular,  possessing  well-defined  fissure  characteristics  to  a 
depth  of  150  to  250  feet,  the  working  vertical  shaft  might  be 
sunk  with  comparative  safety.  No  rule  can  be  laid  down,  how- 
ever, and  the  Engineer  must  always  exercise  his  own  judgment. 

In  the  example  here  given,  we  have  assumed  the  inclined  shaft 
to  be  sunk  to  the  same  depth  as  the  vertical,  for  if  originally 
sunk  to  the  first  cross-cut,  440  feet  (Eig.  106),  it  would  be  car- 
ried down  to  the  point  of  intersection  and  used  for  ventilation. 

The  method  of  surveying  and  plotting  such  a mine  is  simply 
an  application  of  the  principles  already  explained. 

408.  The  calculation  of  ore-reserves  does  not  come  strictly 
within  the  province  of  the  surveyor,  yet  after  completing  the 
survey  and  plot,  he  is  frequently  required  to  make  the  calculation. 
We  will  therefore  consider  methods  of  making  it. 

In  practice  the  methods  employed  are  various.  No  general 
rule  can  be  given,  as  each  expert  has  a system  of  his  own,  and  dif- 
ferent engineers  will  not  agree,  within  wide  limits,  as  to  the 
quantity  of  ore-reserves  in  the  same  mine.  One  may  assume  as 
the  measure  of  the  ore  in  sight  a rectangular  block  limited  by 
the  outcrop,  the  depth  of  the  shaft  or  shafts,  and  the  extreme 
points  of  the  levels,  diminished  by  the  amount  extracted. 
Others,  but  one-half  or  one-third  of  this  quantity.  The  former 
would  be  considered  an  excessive  estimate  in  all  cases.  The  lat- 
ter too  low  when  the  vein  possesses  great  strength  and  regularity, 
though  even  this  estimate  may  be  too  high,  when  the  conditions 
are  the  reverse.  The  surveyor  must  exercise  his  own  judgment, 
exercising  caution,  however,  as  the  calculation  is  an  important 
one. 

Assume  the  development  shown  in  Figs.  104,  105,  106,  and 
let  it  he  required  to  calculate  the  ore-reserves  when  the  hounding 
lines  are  assumed  at  the  extreme  ends  of  the  level  drifts,  or  the 


SEC.  IV.] 


PRACTICAL  APPLICATIONS. 


371 


rectangular  block  ABCD,  Fig.  195,  and  when  the  average  cross- 
section  of  the  vein  is  6 feet,  and  a cubic  foot  of  the  vein  matter 
in  place  weighs  150  lbs. 

Ore  stopes,  or  steps  made  in  mine- workings  by  and  for 
the  extraction  of  ore,  are  generally  very  irregular,  the  repre- 
sentation here  being  an  ideal  one.  Suppose  the  stope-faces  to  be 
11  feet  apart  and  8 feet  high,  and  that  the  inclined  shaft  has 
extracted  10  x 6 ft.  of  vein  matter,  and  the  levels  7 x 6 f t. 

We  see  that  the  inclined  shaft  has  exposed  the  vein  for 
440  + 115  + 115  = G70  ft.;  deducting  say  15  ft.  for  inequality  of 
surface,  we  should  have  a rectangular  block  655  x (400  + 350) 
x 6 in  width  = 2947500  cubic  feet  of  ore : to  be  deducted  from 
this,  we  have 

The  inclined  shaft  655  x 10  x 6 = 39300  cubic  feet. 


1st  level  (150  + 200)  x 7 x 6 

= 14700 

{( 

a 

2d  “ 

260  x 7 X 6 

= 10920 

(( 

a 

3d  “ 

520  x 7 x 6 

= 21840 

{( 

a 

4th  “ 

240  x 7 x 6 

= 10080 

(( 

a 

5th  “ 

345  x 7 X 6 

= 14490 

t( 

a 

6th  “ 

500  x 7 x 6 

= 21000 

n 

a 

Stoped  out  on  1st  level  east,  roughly  estimated,  3400 

a 

a 

it 

“ “ “ west  “ 

6500 

a 

a 

({ 

“ 2d  “ “ 

“ 7000 

a 

a 

« 

“ 3d  “ east  “ 

“ 20000 

a 

a 

ee 

“ 6th  “ west  “ 

“ 12000 

a 

a 

Total,  182000  cu.  ft.  (say). 

Deducting  this  from  2947500,  we  have,  2765500  “ “ 

Dividing  by  13£,  the  number  of  cu.  ft.  required  for  a ton,  and  we 
have  204852  tons  of  ore  in  sight. 

Another  method  of  calculation  is  as  follows:  The  longest 


372 


ELEMENTS  OF  SURVEYING. 


[ROOK  X. 


drift  east  is  400  ft.  and  the  shortest  100  ft.  Assume  the  bounding 
line  in  this  direction  to  be  at  a distance  east  of  the  shaft. 


100  + 


400  — 100 
2 


250  ft. 


The  longest  drift  west  is  350  ft.  and  the  shortest  100  ft.;  take  the 
bounding  line  in  this  direction  at  a distance 


100  + 


350  — 100 

2 “ 


= 225  ft., 


or  the  rectangular  block  abcF,  Fig.  195.  Calculate  the  ore- 
reserves,  when  the  other  data  are  the  same  as  before. 

This  latter  method  is  recommended  by  competent  engineers 
as  the  fairest  and  most  reliable  for  all  parties  concerned. 

409.  Fig.  197  shows  a longitudinal  section,  and  Fig.  198  a 
transverse  section  of  a deposit  mine  with  mill  connections,  the 
mill  to  be  erected  at  the  point  A.  From  a consideration  of  the 
diagram,  it  is  evident  that  the  most  convenient  method  for  the 
transportation  of  the  ore  from  the  mine  to  the  mill  would  be  by 
a tunnel  driven  into  the  mountain,  at  the  end  of  which  is  a bin, 
made  in  the  solid  rock  and  inclined  to  the  tunnel  at  any  convenient 
angle  at  which  ore  will  slide  into  cars;  the  cars  to  be  run  into  the 
tunnel  on  a track  and  directly  under  iron  doors  which  are  worked 
by  rack  and  pinion.  The  bin  is  to  connect  with  the  ore-chamber 
by  a chute  inclined  at  an  angle  of  45°,  as  shown  in  the  diagram. 

The  lower  or  mill  tunnel  should  have  a slope  of  2 inches  in 
10  leet,  so  that  the  loaded  cars  would  descend  by  the  force  of 
gravity,  the  last  car  in  a train  having  a brake  with  which  to 
regulate  the  speed.  The  chute  should  be  12  to  15  feet  from  the 
edge  of  the  tunnel,  to  admit  of  constructing  the  inclined  bin  for 
the  discharge  of  the  ore  into  the  cars.  The  point  in  the  ore- 
chamber,  at  which  it  is  desired  to  sink  the  chute,  and  the  mouth 
of  the  lower  tunnel  being  solected,  drive  a peg  to  the  centre  point 


Transverse  Section 
Chute 


SEC.  IV.] 


practical  applications. 


m 


374 


ELEMENTS  OF  SURVEYING. 


[ROOK  X. 


of  the  proposed  tunnel  floor  and  drive  a nail  in  the  peg,  and 
repeat  the  operation  at  the  point  where  the  chute  is  to  be  sunk. 
Now  make  a careful  traverse  between  these  points  ; the  direction 
of  a line  which  will  run*  from  the  mouth  of  the  tunnel  directly 
under  the  point  selected  for  the  chute  can  now  be  found,  as 
explained  in  the  section  on  traversing,  and  the  course  that  will 
carry  the  tunnel  12  or  15  feet  from  the  bottom  of  the  chute  may 
be  determined.  In  driving  the  tunnel,  holes  should  be  drilled  in 
the  roof  and  wooden  spuds  driven  in  on  which  to  hang  plumb-bobs, 
the  surveyor  using  great  care  to  have  the  plumb-bobs  suspended 
in  the  proper  course  as  a guide  to  the  miners.  In  starting  the 
chute,  a large  wooden  triangle  should  be  made,  one  of  the  angles 
of  which  is  the  same  as  the  angle  of  the  chute,  to  be  used 
by  the  miners  as  a guide,  until  sufficient  depth  is  attained  to 
hang,  in  the  proper  line,  plumb-bobs,  the  points  of  which  are  on 
the  required  angle. 


APPENDIX  A. 


TH  E SOLAR  COM  PASS. 

( With  some  omissions,  from  Messrs.  W.  and  L.  E.  Gurley’s  Manual  of  Engi- 
neering and  Surveying  Instruments,  21ftli  Edition,  1883.) 

This  instrument,  so  ingeniously  contrived  for  readily  determining  a true 
meridian  or  north  and  south  line,  was  invented  by  William  A.  Burt,  of  Mich- 
igan, and  patented  by  him  in  1836. 

It  has  since  come  into  general  use  in  the  surveys  of  U.  S.  public  lands, 
the  principal  lines  of  which  are  required  to  be  run  with  reference  to  the  true 
meridian. 

The  arrangement  of  its  sockets  and  plates  is  similar  to  that  of  the  Survey- 
ors’ Transit,  except  that  the  sight  vanes  are  attached  to  the  under  plate  or 
limb,  and  this  revolves  around  the  upper  or  vernier  plate  on  which  the  solar 
apparatus  is  placed. 

The  limb  is  divided  to  half  degrees,  is  figured  in  two  rows,  and  reads  by 
the  two  opposite  verniers  to  single  minutes. 

The  Solar  Apparatus.— The  Solar  Apparatus  is  seen.  Fig.  1,  in 
the  place  of  the  needle,  and  in  fact  operates  as  its  substitute  in  the  field. 

It  consists  mainly  of  three  arcs  of  circles,  by  which  can  be  set  off  the  lati- 
tude of  a place,  the  declination  of  the  sun,  and  the  hour  of  the  day. 

These  arcs,  designated  in  the  cut  by  the  letters  a,  b,  and  c,  are  therefore 
termed  the  latitude,  the  declination,  and  the  hour  arcs  respectively. 

The  Latitude  Arc,  a,  has  its  centre  of  motion  in  two  pivots,  one  of 
which  is  seen  at  d,  the  other  is  concealed  in  the  cut. 

It  is  moved  either  up  or  down  within  a hollow  arc,  seen  in  the  cut,  by  a 
tangent  screw  at /,  and  is  securely  fastened  in  any  position  by  a clamp  screw 

The  Latitude  arc  is  graduated  to  quarter  degrees,  and  reads  by  its  vernier, 
e,  to  single  minutes  ; it  has  a range  of  about  thirty-five  degrees,  so  as  to  be 
adjustable  to  the  latitude  of  any  place  in  the  United  States. 

The  Declination  Arc,  b,  is  also  graduated  to  quarter  degrees,  and 
has  a range  of  about  twenty-eight  degrees. 

Its  vernier,  v , reading  to  single  minutes,  is  fixed  to  a movable  arm,  h,  hav- 
ing its  centre  of  motion  at  the  end  of  the  declination  arc  at  g ; the  arm  is 
moved  over  the  surface  of  the  declination  arc,  and  its  vernier  set  to  any  read- 
ing by  turning  the  head  of  the  tangent  screw  k.  It  is  also  securely  clamped 
in  any  position  by  a screw,  concealed  in  the  engraving. 

Solar  Lenses  and  Lines. — At  each  end  of  the  arm,  h,  is  a rectangu- 
lar block  of  brass,  in  which  is  set  a small  convex  lens,  having  its  focus  on  the 


ELEMENTS  OF  SURVEYING. 


9 


fj 


[APP.  A. 


surface  of  a little  silver  plate  A,  Fig.  2,  fastened  by  screws  to  the  inside  of 

the  opposite  block. 


£ 


On  the  surface  of  the  plate  are  marked  two  sets  of  H~ro~n 

lines  intersecting  each  other  at  right  angles  ; of  these  b b <3  ' \ . 

are  termed  the  hour  lines,  and  c c the  equatorial  lines,  as  (7\~" 

having  reference  respectively  to  the  hour  of  the  day  and 1 

the  position  of  the  sun  in  relation  to  the  equator.  Fig.  2. 

In  Fig.  1 the  equatorial  lines  are  those  on  the  lower  block,  parallel  to  the 
surface  of  the  hour  arc  c ; the  hour  lines  are  of  course  those  at  right  angles 
to  the  first. 


0 - l 

0”1 

is  &I 
1 

Equatorial  Sights  On  the  top  of  each  of  the  rectangular  blocks 
is  seen  a little  sighting-piece,  termed  the  equatorial  sight,  fastened  to  the 
block  by  u small  milled  head  screw,  so  as  to  be  detached  at  pleasure. 


APP.  A.] 


THE  SOLAR  COMPASS. 


3 


They  are  used,  as  will  be  explained  hereafter,  in  adjusting  the  different 
parts  of  the  solar  apparatus. 

The  Hour  Arc,  c,  is  supported  by  the  two  pivots  of  the  latitude  arc 
already  spoken  of,  and  is  also  connected  with  that  arc  by  a curved  arm,  as 
shown  in  the  figure. 

The  hour  arc  has  a range  of  about  120°,  is  divided  to  half  degrees,  and 
figured  in  two  series  ; designating  both  the  hours  and  the  degrees,  the  middle 
division  being  marked  12  and  90  on  either  side  of  the  graduated  lines. 

The  Polar  Axis. — Through  the  centre  of  the  hour  arc  passes  a hollow 
socket,  p,  containing  the  spindle  of  the  declination  arc,  by  means  of  which 
this  arc  can  be  moved  from  side  to  side  over  the  surface  of  the  hour  arc,  or 
turned  completely  round,  as  may  be  required. 

The  hour  arc  is  read  by  the  lower  edge  of  the  graduated  side  of  the  decli- 
nation arc. 

The  axis  of  the  declination  arc,  or  indeed  the  whole  socket  p,  is  appropri- 
ately termed  the  polar  axis. 

The  Adjuster. — Besides  the  parts  shown  in  the  cut,  there  is  also  an 
arm  used  in  the  adjustment  of  the  instrument  as  described  hereafter,  but  laid 
aside  in  the  box  when  that  is  effected. 

The  parts  just  described  constitute  properly  the  solar  apparatus. 

Besides  these,  however,  are  seen  the  needle-box,  n,  with  its  arc  and  tan- 
gent screw,  t,  and  the  spirit  levels,  for  bringing  the  whole  instrument  to  a 
horizontal  position. 

The  Needle-Box,  n,  has  an  arc  of  about  36°  in  extent,  divided  to  half 
degrees,  and  figured  from  the  centre  or  zero  mark  on  either  side. 

The  needle  is  raised  or  lowered  by  a lever  shown  in  the  cut. 

The  needle-box  is  attached  by  a projecting  arm  to  a tangent- screw,  t,  by 
which  it  is  moved  about  its  centre,  and  its  needle  set  to  any  variation. 

This  variation  is  also  read  off  by  the  vernier  on  the  end  of  the  projecting 
arm,  reading  to  three  minutes  a graduated  arc,  attached  to  the  plate  of  the 
compass. 

The  Levels  seen  with  the  solar  apparatus  have  ground  glass  vials,  and 
are  adjustable  at  their  ends  like  those  of  other  instruments. 

The  edge  of  the  circular  plate  on  which  the  solar  work  is  placed,  is  divided 
and  figured  at  intervals  of  ten  degrees,  and  numbered,  as  shown,  from  0 to  90 
on  each  side  of  the  line  of  sight. 

These  graduations  are  used  in  connection  with  a little  brass  pin,  seen  in 
the  centre  of  the  plate,  to  obtain  approximate  bearings  of  lines,  which  are 
not  important  enough  to  require  a close  observation. 

Lines  Of  Refraction  —The  inside  faces  of  the  sights  are  also  gradu- 
ated and  figured,  to  indicate  the  amount  of  refraction  to  be  allowed  when  the 
sun  is  near  the  horizon.  These  are  not  shown  in  the  cut. 

Principles  of  the  Solar  Compass. — The  interval  between  two 
equatorial  lines  c c , in  Fig.  2,  as  well  as  between  the  hour  lines  b b,  is  just 
sufficient  to  include  the  circular  image  of  the  sun  as  formed  by  the  solar  lens 
on  the  opposite  end  of  the  revolving  arm  h,  Fig.  1. 

When,  therefore,  the  instrument  is  made  perfectly  horizontal,  the  equa- 


4 


ELEMENTS  OF  SURVEYING. 


[APP.  A. 


torial  lines  and  the  opposite  lenses  being  accurately  adjusted  to  each  other 
by  a previous  operation,  and  the  sun’s  image  brought  within  the  equatorial 
lines,  his  position  in  the  heavens,  with  reference  to  the  horizon,  will  be  de- 
fined with  precision. 

Suppose  the  observation  to  be  made  at  the  time  of  one  of  the  equinoxes  ; 
the  arm  h,  set  at  zero  on  the  declination  arc  b,  and  the  polar  axis  p placed 
exactly  parallel  to  the  axis  of  the  earth. 

Then  the  motion  of  the  arm  h,  if  revolved  on  the  spindle  of  the  declination 
arc  around  the  hour  circle  c,  will  exactly  correspond  with  the  motion  of  the 
sun  in  the  heavens,  on  the  given  day  and  at  the  place  of  observation  ; so  that 
if  the  sun’s  image  was  brought  between  the  lines  c c,  in  the  morning,  it 
would  continue  in  the  same  position,  passing  neither  above  nor  below  the 
lines,  as  the  arm  was  made  to  revolve  in  imitation  of  the  motion  of  the  sun 
about  the  earth. 

In  the  morning,  as  the  sun  rises  from  the  horizon,  the  arm  h will  be  in  a 
position  nearly  at  right  angles  to  that  shown  in  the  cut,  the  lens  being  turned 
towards  the  sun,  and  the  silver  plate  on  which  his  image  is  thrown  directly 
opposite. 

As  the  sun  ascends,  the  arm  must  be  moved  around,  until,  when  he  has 
reached  the  meridian,  the  graduated  side  of  the  declination  arc  will  indicate 
12  on  the  hour  circle,  and  the  arm  h,  the  declination  arc  b,  and  the  latitude 
arc  a,  will  be  in  the  same  plane. 

As  the  sun  declines  from  the  meridian  the  arm  h must  be  moved  in  the 
same  direction,  until  at  sunset  its  position  will  be  the  exact  reverse  of  that  it 
occupied  in  the  morning. 

Allowance  for  Declination. — Let  us  now  suppose  the  observation 
made  when  the  sun  has  passed  the  equinoctial  point,  and  when  his  position 
is  affected  by  declination. 

By  referring  to  the  Almanac,  and  setting  off  on  the  arc  his  declination  for 
the  given  day  and  hour,  we  are  still  able  to  determine  his  position  with  the 
same  certainty  as  if  he  remained  on  the  equator. 

When  the  sun’s  declination  is  south,  that  is,  from  the  22d  of  September 
to  the  20tli  of  March  in  each  year,  the  arc  b is  turned  toward  the  plates  of  the 
compass,  as  shown  in  the  engraving,  and  the  solar  lens,  o,  with  the  silver 
plate  opposite,  are  made  use  of  in  the  surveys. 

The  remainder  of  the  year,  the  arc  is  turned  from  the  plates,  and  tlia 
other  lens  and  plate  employed. 

When  the  Solar  Compass  is  accurately  adjusted,  and  its  plates  made  per- 
fectly horizontal,  the  latitude  of  the  place,  and  the  declination  of  the  sun  for 
the  given  day  and  hour,  being  also  set  off  on  the  respective  arcs,  the  image 
of  the  sun  cannot  be  brought  between  the  equatorial  lines  until  the  polar  axis 
is  placed  in  the  plane  of  the  meridian  of  the  place,  or  in  a position  parallel • to 
Ihe  axis  of  the  earth.  The  slightest  deviation  from  this  position  will  cause 
the  image  to  pass  above  or  below  the  lines,  and  thus  discover  the  error. 

We  thus,  from  the  position  of  the  sun  in  the  solar  system,  obtain  a certain 
direction  absolutely  unchangeable,  from  which  to  run  our  lines,  and  measure 
the  horizontal  angles  required. 

This  simple  principle  is  not  only  the  basis  of  the  construction  of  the  Solar 
Compass,  but  the  sole  cause  of  its  superiority  to  the  ordinary  or  magnetic  in- 
strument. For  in  a needle  instrument  tho  accuracy  of  the  horizontal  angles 


APP.  A.] 


THE  SOLAR  COMPASS. 


indicated,  and  therefore  of  all  tlie  observations  made,  depends  upon  u the  deli- 
cacy of  the  needle,  and  the  constancy  with  which  it  assumes  a certain  direc- 
tion, termed  the  magnetic  meridian.” 

The  principal  causes  of  error  in  the  needle,  briefly  stated,  are  the  dulling 
of  the  pivot,  the  loss  of  polarity  in  the  needle,  the  influence  of  local  attraction, 
and  the  effect  of  the  sun’s  rays,  producing  the  diurnal  variation. 

From  all  these  imperfections  the  solar  instrument  is  free. 

The  sights  and  the  graduated  limb  being  adjusted  to  the  solar  apparatus, 
and  the  latitude  of  the  place  and  the  declination  of  the  sun  also  set  off  upon 
the  respective  arcs,  we  are  able  not  only  to  run  the  true  meridian,  or  a due 
east  and  west  course,  but  also  to  set  off  the  horizontal  angles  with  minuteness 
and  accuracy  from  a direction  which  never  changes,  and  is  unaffected  by  at- 
traction of  any  kind. 

To  Adjust  the, Solar  Compass.— The  adjustments  of  this  in- 
strument, with  which  the  surveyor  will  have  to  do,  are  few  in  number,  and 
will  now  be  given  in  order. 

1st.  To  Adjust  the  Levels. — Proceed  precisely  as  directed  in  the 
account  of  the  other  instruments  described,  by  bringing  the  bubbles  into  the 
centre  of  the  tubes  by  the  leveling  screws  of  the  tripod,  and  then  reversing 
the  instrument  upon  its  spindle,  and  raising  or  lowering  the  ends  of  the  tubes 
until  the  bubbles  will  remain  in  the  centre  during  a complete  revolution  of 
the  instrument. 

2d.  To  Adjust  the  Equatorial  Lines  and  Solar  Lenses.— 

First  detach  the  arm  h from  the  declination  arc  by  withdrawing  the  screws 
shown  in  the  cut  from  the  ends  of  the  posts  of  the  tangent-screw  k,  and  also 
the  clamp-screw,  and  the  conical  pivot  with  its  small  screws  by  which  the  arm 
and  declination  arc  are  connected. 

The  arm  h being  thus  removed,  attach  the  adjuster  in  its  place  by  replac- 
ing the  conical  pivot  and  screws,  and  insert  the  clamp-screw  so  as  to  clamp 
the  adjuster  at  any  point  on  the  declination  arc. 

Now  level  the  instrument,  place  the  arm  h on  the  adjuster,  with  the  same 
side  resting  against  the  surface  of  the  declination  areas  before  it  was  detached. 
Turn  the  instrument  on  its  spindle  so  as  to  bring  the  solar  lens  to  be  adjusted 
in  the  direction  of  the  sun,  and  raise  or  lower  the  adjuster  on  the  declination 
arc,  until  it  can  be  clamped  in  such  a position  as  to  bring  the  sun’s  image  as 
near  as  may  be  between  the  equatorial  lines  on  the  opposite  silver  plate,  and 
bring  the  image  precisely  into  position  by  the  tangent  of  the  latitude  arc  or 
the  leveling-screws  of  the  tripod.  Then  carefully  turn  the  arm  half  way 
over,  until  it  rests  upon  the  adjuster  by  the  opposite  faces  of  the  rectangular 
blocks,  and  again  observe  the  position  of  the  sun’s  image. 

If  it  remains  between  the  lines  as  before,  the  lens  and  plate  are  in  adjust- 
ment ; if  not,  loosen  the  three  screws  which  confine  the  plate  to  the  block, 
and  move  the  plate  under  their  heads,  until  one-half  the  error  in  the  position 
of  the  sun’s  image  is  removed. 

Again  bring  the  image  between  the  lines,  and  repeat  the  operation  until  it 
will  remain  in  the  same  situation,  in  both  positions  of  the  arm,  when  the  ad- 
justment will  be  completed. 

To  adjust  the  other  lens  and  plate,  reverse  the  arm  end  for  end  on  the  ad 


ELEMENTS  OP  SURVEYING. 


G 


[APP.  A. 


juster,  and  proceed  precisely  as  in  the  former  case,  until  the  same  result  is 
attained. 

In  tightening  the  screws  over  the  silver  plate,  care  must  be  taken  not  to 
move  the  plate. 

This  adjustment  now  being  complete,  the  adjuster  should  be  removed, 
and  the  arm  h , with  its  attachments,  replaced  as  before. 

3d.  To  Adjust  the  Vernier  of  the  Declination  Arc.— Hav 

ing  leveled  the  instrument,  and  turned  its  lens  in  the  direction  of  the  sun, 
clamp  to  the  spindle,  and  set  the  vernier  v,  of  the  declination  arc,  at  zero,  by 
means  of  the  tangent-screw  at  k,  and  clamp  to  the  arc. 

See  that  the  spindle  moves  easily  and  yet  truly  in  the  socket,  or  polar 
axis,  and  raise  or  lower  the  latitude  arc  by  turning  the  tangent-screw/,  until 
the  sun’s  image  is  brought  between  the  equatorial  lines  on  one  of  the  plates. 
Clamp  the  latitude  arc  by  the  screw,  and  bring  the  image  precisely  into  posi- 
tion by  the  leveling-screws  of  the  tripod  or  socket,  and  without  disturbing 
the  instrument,  carefully  revolve  the  arm  h,  until  the  opposite  lens  and  plate 
are  brought  in  the  direction  of  the  sun,  and  note  if  the  sun’s  image  comes  be- 
tween the  lines  as  before. 

If  it  does,  there  is  no  index  error  of  the  declination  arc  ; if  not,  with  the 
tangent-screw  k,  move  the  arm  until  the  sun’s  image  passes  over  half  the 
error  ; again  bring  the  image  between  the  lines,  and  repeat  the  operation  as 
before,  until  the  image  will  occupy  the  same  position  on  both  the  plates. 

We  shall  now  find,  however,  that  the  zero  marks  on  the  arc  and  the  ver- 
nier do  not  correspond,  and  to  remedy  this  error,  the  little  flat-head  screws 
above  the  vernier  must  be  loosened  until  it  can  be  moved  so  as  to  make  the 
zeros  coincide,  when  the  operation  will  be  completed. 

4th.  To  Adjust  the  Solar  Apparatus  to  the  Compass 

Sights. — First,  level  the  instrument,  and  with  the  clamp  and  tangent- 
screws  set  the  main  plate  at  90°  by  the  verniers  and  horizontal  limb.  Then 
remove  the  clamp  screw,  and  raise  the  latitude  arc  until  the  polar  axis  is  by 
estimation  very  nearly  horizontal,  and  if  necessary,  tighten  the  screws  on  the 
pivots  of  the  arc,  so  as  to  retain  it  in  this  position. 

Fix  the  vernier  of  the  declination  arc  at  zero,  and  direct  the  equatorial 
sights  to  some  distant  and  well-marked  object,  and  observe  the  same  through 
the  compass  sights.  If  the  same  object  is  seen  through  both,  and  the  ver- 
niers read  to  90°  on  the  limb,  the  adjustment  is  complete  ; if  not,  the  correc- 
tion must  be  made  by  moving  the  sights  or  changing  the  position  of  the. 
verniers. 

It  should  be  remarked  that  as  the  solar  work  is  attached  permanently  to 
the  sockets,  and  this  adjustment  is  made  by  the  maker,  it  will  need  no  fur- 
ther attention  at  the  hands  of  the  surveyor  except  in  case  of  serious  accidents. 

The  other  adjustments  are  of  course  also  made  in  the  process  of  finishing 
the  instrument,  and  are  liable  to  very  little  derangement  in  the  ordinary  use 
of  the  Solar  Compass. 

To  Use  the  Solar  Compass. — Before  this  instrument  can  be  used 
at  any  given  place,  it  js  necessary  to  set  off  upon  its  arcs  both  the  declination 
of  the  sun  as  affected  by  its  refraction  for  the  given  day  and  hour,  and  the 
latitude  of  tin1  place  where  the  observation  is  made. 


APP.  A.] 


THE  SOLAR  COMPASS. 


7 


To  Set  Off  the  Declination. — The  declination  of  the  sun,  given 
in  the  epliemeris  of  the  Nautical  Almanac  from  year  to  year,  is  calculated  for 
apparent  noon  at  Greenwich,  England. 

To  determine  it  for  any  other  hour  at  a place  in  the  U.  S.,  reference  must 
be  had,  not  only  to  the  difference  of  time  arising  from  the  longitude,  but  also 
to  the  change  of  declination  from  day  to  day. 

The  longitude  of  the  place,  and  therefore  its  difference  in  time,  if  not 
given  directly  in  the  tables  of  the  Almanac,  can  be  ascertained  very  nearly 
by  reference  to  that  of  other  places  given,  which  are  situated  on,  or  very 
nearly  on,  the  same  meridian. 

it  is  the  practice  of  surveyors  in  the  States  east  of  the  Mississippi  to  allow 
a difference  of  six  hours  for  the  difference  in  longitude,  calling  the  declination 
given  in  the  Almanac  for  12  M.,  that  of  6 A.  M.,  at  the  place  of  observation. 

Beyond  the  meridian  of  Santa  Fe,  the  allowance  would  be  about  seven 
hours,  and  in  California,  Oregon,  and  Washington  Territory  about  eight  hours. 

Having  thus  the  difference  of  time,  we  very  readily  obtain  the  declination 
for  a certain  hour  in  the  morning,  which  would  be  earlier  or  later  as  the  lon- 
gitude was  greater  or  less,  and  the  same  as  that  of  apparent  noon  at  Green- 
wich on  the  given  day.  Thus,  suppose  the  observation  made  at  a place,  say, 
five  hours  later  than  Greenwich,  then  the  declination  given  in  the  Almanac 
for  the  given  day  at  noon,  affected  by  the  refraction,  would  be  the  declination 
at  the  place  of  observation  for  7 o’clock  A.  M.  ; this  gives  us  the  starting- 
point. 

To  obtain  the  declination  for  the  other  hours  of  the  day,  take  from  the 
Almanac  the  declination  for  apparent  noon  of  the  given  day,  and,  as  the  dec- 
lination is  increasing  or  decreasing,  add  to  or  subtract  from  the  declination  of 
the  first  hour  the  difference  for  one  hour  as  given  in  the  ephemeris,  which 
will  give,  when  affected  by  the  refraction,  the  declination  for  the  succeeding 
hour  ; and  proceed  thus  in  making  a table  of  the  declination  for  every  hour 
of  the  day. 

To  Set  OIF  the  Latitude. — Find  the  declination  of  the  sun  for  the 
given  day  at  noon,  at  the  place  of  observation  as  just  described,  and  with  the 
tangent-screw  set  it  off  upon  the  declination  arc,  and  clamp  the  arm  firmly  to 
the  arc. 

Observe  in  the  Almanac  the  equation  of  time  for  the  given  day,  in  order  to 
know  about  the  time  the  sun  will  reach  the  meridian. 

Then,  about  fifteen  or  twenty  minutes  before  this  time,  set  up  the  instru- 
ment, level  it  carefully,  fix  the  divided  surface  of  the  declination  arc  at  12  on 
the  hour  circle,  and  turn  the  instrument  upon  its  spindle  until  the  solar  lens 
is  brought  into  the  direction  of  the  sun. 

Loosen  the  clamp-screw  of  the  latitude  arc,  and  with  the  tangent-screw 
raise  or  lower  this  arc  until  the  image  of  the  sun  is  brought  precisely  between 
the  equatorial  lines,  and  turn  the  instrument  from  time  to  time  so  as  to  keep 
the  image  also  between  the  hour  lines  on  the  plate. 

As  the  sun  ascends,  its  image  will  move  below  the  lines,  and  the  arc  must 
be  moved  to  follow  it.  Continue  thus,  keeping  it  between  the  two  sets  of 
lines  until  its  image  begins  to  pass  above  the  equatorial  lines,  which  is  also 
the  moment  of  its  passing  the  meridian. 

Now  read  off  the  vernier  of  the  arc,  and  we  have  the  latitude  of  the  place. 


8 ELEMENTS  OF  SURVEYING.  [API*.  A. 

which  is  always  to  be  set  off  on  the  arc  when  the  compass  is  used  at  the 
given  place. 

It  is  the  practice  of  surveyors  using  the  Solar  Compass  to  set  off,  in  the 
manner  just  described,  the  latitude  of  the  point  where  the  survey  begins,  and 
to  repeat  the  observation  and  correction  of  the  latitude  arc  every  day  when 
the  weather  is  favorable,  there  being  also  nearly  an  hour  at  mid-day  when 
the  sun  is  so  near  the  meridian  as  not  to  give  the  direction  of  lines  with  the 
certainty  required. 

To  Run  Lines  with  the  Solar  Compass.— Having  set  off  in 
the  manner  just  given  the  latitude  and  declination  upon  their  respective 
arcs,  the  instrument  being  also  in  adjustment,  the  surveyor  is  ready  to  run 
lines  by  the  sun. 

To  do  this,  the  instrument  is  set  over  the  station  and  carefully  leveled, 
the  plates  clamped  at  zero  on  the  horizontal  limb,  and  the  sights  directed 
north  and  south,  the  direction  being  given,  when  unknown,  approximately  by 
the  needle. 

The  solar  lens  is  then  turned  to  the  sun,  and  with  one  hand  on  the  in- 
strument, and  the  other  on  the  revolving  arm,  both  are  moved  from  side  to 
side,  until  the  sun’s  image  is  made  to  appear  on  the  silver  plate  ; when  by 
carefully  continuing  the  operation,  it  may  be  brought  precisely  between  the 
equatorial  lines. 

Allowance  being  made  for  refraction,  the  line  of  sights  will  indicate  the 
true  meridian  ; the  observation  may  now  be  made,  and  the  flag-man  put  in 
position. 

When  a due  east  and  west  line  is  to  be  run,  the  verniers  of  the  horizontal 
limb  are  set  at  90°,  and  the  sun’s  image  kept  between  the  lines  as  before. 

The  Solar  Compass  being  so  constructed  that  when  the  sun’s  image  is  in 
position  the  limb  must  be  clamped  at  0 in  order  to  run  a true  meridian  line, 
it  will  be  evident  that  the  bearing  of  any  line  from  the  meridian  may  be  read 
by  the  verniers  of  the  limb,  precisely  as  in  the  ordinary  magnetic  compass 
the  bearing  of  lines  are  read  from  the  ends  of  the  needle. 

Use  of  the  Needle. — In  running  lines,  the  magnetic  needle  is  always 
kept  with  the  sun  ; that  is,  the  point  of  the  needle  is  made  to  indicate  0 on 
the  arc  of  the  compass-box,  by  turning  the  tangent-screw  connected  with  its 
arm  on  the  opposite  side  of  the  plate.  By  this  means  the  lines  can  be  run  by 
the  needle  alone  in  case  of  the  temporary  disappearance  of  the  sun ; but,  of 
course,  in  such  cases  the  surveyor  must  be  sure  that  no  local  attraction  is 
exerted. 

The  variation  of  the  needle,  which  is  noted  at  every  station,  is  read  off  in 
degrees  and  minutes  on  the  arc,  by  the  edge  of  which  the  vernier  of  the 
needle-box  moves. 

Allowance  for  the  Earth’s  Curvature.— When  long  lines  are 
run  by  the  Solar  Compass,  cither  by  the  true  meridian,  or  due  east  and  west, 
allowance  must  be  made  for  the  curvature  of  the  earth. 

Thus,  in  running  north  or  south,  the  latitude  changes  about  one  minute 
for  every  distance  of  92  chains  .30  links,  and  tlio  side  of  a township  requires 
a change  on  the  latitude  arc  of  5'  12",  the  township,  of  course,  being  six  miles 
square. 


APP.  A.] 


THE  SOLAR  COMPASS. 


0 


This  allowance  is  of  constant  use  where  the  surveyor  fails  to  get  an  obser- 
vation on  the  sun  at  noon,  and  is  a very  close  approximation  to  the  truth. 

In  running  due  east  and  west,  as  in  tracing  the  standard  parallels  of  lati- 
tude, the  sights  are  set  at  90°  on  the  limb,  and  the  line  is  run  at  right  angles  to 
the  meridian. 

If  no  allowance  were  made  for  the  earth’s  curvature,  these  lines  would,  if 
sufficiently  produced,  reach  the  equator,  to  which  they  are  constantly  tending. 

Of  course,  in  running  short  lines  either  east  or  west,  the  variation  from 
the  parallel  would  be  so  small  as  to  be  of  no  practical  importance  ; but  when 
long  sights  are  taken,  the  correction  should  be  made  by  taking  fore  and  back 
sights  at  every  station,  noticing  the  error  on  the  back  sight,  and  setting  off 
one-half  of  it  on  the  fore  sight  on  the  side  towards  the  pole. 

Time  Of  Day  by  the  Sun. — The  time  of  day  is  best  ascertained  by 
the  Solar  Compass  when  the  sun  is  on  the  meridian,  as  at  the  time  of  making 
the  observation  for  latitude. 

The  time  thus  given  is  that  of  apparent  noon,  and  can  be  reduced  to  mean 
time  by  merely  applying  the  equation  of  time  as  directed  in  the  Almanac,  and 
adding  or  subtracting  as  the  sun  is  slow  or  fast. 

The  time,  of  course,  can  also  be  taken  before  or  after  noon,  by  bringing 
the  sun’s  image  between  the  hour  lines,  and  noticing  the  position  of  the 
divided  edge  of  the  revolving  arm,  with  reference  to  the  graduations  of  the 
hour  circle,  allowing  four  minutes  of  time  for  each  degree  of  the  arc,  and  thus 
obtaining  apparent  time,  which  must  be  corrected  by  the  equation  of  time  as 
just  decribed. 

Caution  as  to  the  False  Image.— In  using  the  compass  upon  the 
sun,  if  the  revolving  arm  be  turned  a little  one  side  of  its  proper  position,  a 
false  or  reflected  image  of  the  sun  will  appear  on  the  silver  plate  in  nearly 
the  same  place  as  that  occupied  by  the  true  one.  It  is  caused  by  the  reflec- 
tion of  the  true  image  from  the  surface  of  the  arm,  and  is  a fruitful  source 
of  error  to  the  inexperienced  surveyor.  It  can,  however,  be  readily  distin- 
guished from  the  real  image  by  being  much  less  bright,  and  not  so  clearly 
defined. 

Approximate  Bearings. — When  the  bearings  of  lines,  such  as  the 
course  of  a stream,  or  the  boundaries  of  a forest,  are  not  desired  with  the  cer 
tainty  given  by  the  verniers  and  horizontal  limb,  a rough  approximation  of 
the  angle  they  make  with  the  true  meridian  is  obtained  by  the  divisions  on 
the  outside  of  the  circular  plate, 

In  this  operation,  a pencil,  or  thin  straight  edge  of  any  sort,  is  held  per- 
pendicularly against  the  circular  edge  of  the  plate,  and  moved  around  until  it 
is  in  range  with  the  eye,  the  brass  centre-pin,  and  the  object  observed. 

The  bearing  of  the  line  is  then  read  off  at  the  point  where  the  pencil  is 
placed. 

Time  for  Using  the  Solar  Compass.— The  Solar  Compass,  like 
the  ordinary  instrument,  can  be  used  at  all  seasons  of  the  year,  the  most  favor- 
able time  being,  of  course,  in  the  summer,  when  the  declination  is  north,  and 
the  days  are  long,  and  more  generally  fair. 

It  is  best  not  to  take  the  sun  at  morning  and  evening,  when  it  is  within 


10 


ELEMENTS  OP  SURVEYING. 


[APP.  A. 


half  an  hour  of  the  horizon,  nor  at  noon,  for  about  the  same  interval,  before 
aud  after  it  passes  the  meridian. 

Allowance  for  Refraction. — Tile  proper  allowance  to  be  made  for 
refraction  in  setting  off  the  declination  of  the  sun  upon  the  Solar  Compass  has 
long  been  a source  of  perplexity  to  the  surveyor  ; we  have,  accordingly,  given 
the  subject  a good  deal  of  attention,  and  here  publish  a table,  by  which  the 
amount  of  refraction  for  any  hour  of  any  day  of  the  year  can  be  ascertained, 
and  set  off  with  a degree  of  accuracy  which  is  all  that  can  be  desired. 

A TABLE  OP  MEAN  REFRACTIONS  IN  DECLINATION. 

To  apply  on  the  declination  arc  of  Solar  Attachment  of  either  Compasses 
or  Transits. 

Computed  by  Edward  W.  Arms,  C.  E.,  for  W.  & L.  E.  Gurley,  Troy, 
N.  Y. 


H 

P 

3 

< 

g 

DECLIN AT 

IONS 

For  Latitude  30'. 

P 

O 

B 

+ 20° 

+ 15° 

+ 10° 

+ 5° 

0° 

— 5° 

—10° 

— 15c 

l 

O 

o 

Oh. 

10" 

15" 

21" 

27" 

33" 

40" 

48" 

57" 

1 '08" 

2 

14 

19 

25 

31 

38 

46 

54 

1'05 

1 18 

1 3 

20 

26 

32 

39 

47 

55 

106 

1 19 

1 36 

4 

32 

39 

46 

52 

1'06 

119 

135 

1 57 

2 29 

5 

l'OO 

110 

1'24 

1'52 

2 07 

2 44 

3 46 

5 43 

13  06 

For  Latitude  32 

0 30'. 

Oh. 

13" 

18" 

24" 

30" 

36" 

44" 

52" 

1'02" 

114" 

2 

17 

22 

28 

35 

42 

50 

l'OO 

1 11 

1 26 

3 

23 

29 

35 

43 

51 

l'Ol 

1 13 

1 28 

1 47 

4 

35 

43 

51 

l'Ol 

113 

1 27 

1 46 

2 13 

2 54 

5 

1'03 

115 

1'31 

1 53 

2 20 

3 05 

4 25 

7 36 

For  Latitude  35 f 

> 

Oh. 

15" 

21" 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

2 

20 

25 

32 

38 

46 

55 

1 05 

1 18 

1 35 

o 

26 

33 

39 

47 

56 

1'07 

1 21 

1 38 

2 00 

4 

39 

47 

56 

1'07 

1'20 

1 36 

1 59 

2 32 

3 25 

5 

1'07 

1'20 

1'38 

2 00 

2 34 

3 29 

5 14 

10  16 

For  Latitude  37° 

30'. 

Oh.  1 

18" 

24" 

30" 

36" 

44" 

52" 

1'02" 

114" 

1'29" 

o | 

22 

28 

35 

42 

50 

l'OO 

1 12 

1 26 

1 45 

3 

29 

36 

43 

52 

1'02 

1 14 

1 29 

1 49 

2 16 

4 

43 

51 

I'Ol 

113 

1 27 

1 49 

2 14 

2 54 

4 05 

5 

I'll 

1'26 

1 54 

2 10 

2 49 

3 55 

6 15 

14  58 

APP.  A.] 


THE  SOLAR  COMPASS. 


11 


1 a 
a 
o 

DECLINATIONS. 

◄ 

1 

For  Latitude  40°. 

o 

W 

+ 20° 

+ 15° 

+ 10° 

+ 5° 

0° 

-5° 

1 

i 1 

o 

o 

-15° 

— 20° 

Oh. 

21" 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'33B 

2 

25 

32 

39 

46 

52 

1'06 

1 19 

1 35 

1 57 

3 

33 

40 

48 

57 

ros 

1 21 

1 38 

2 02 

2 36 

4 

4? 

55 

1'06 

119 

1 36 

1 58 

2 30 

3 21 

4 59 

5 

1'  15 

1'31 

1 51 

2 20 

3 05 

4 25 

7 34 

25  18 

For  Latitude  42°  30'. 

Oh. 

24" 

30" 

36" 

44" 

52" 

1'02" 

114" 

1'29" 

1'49" 

2 

28 

35 

39 

50 

l'OO 

1 12 

1 26 

1 45 

2 11 

3 

36 

43 

52 

102 

1 13 

1 29 

1 49 

2 17 

2 59 

4 

50 

l'OO 

I'll 

1 26 

1 44 

2 10 

2 49 

3 55 

6 16 

5 

1'16 

1 36 

1 58 

2 30 

3 22 

5 00 

9 24 

For  Latitude  45°. 

Oh. 

27" 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'39" 

2'02" 

2 

32 

39 

46 

52 

1'06 

1 19 

1 35 

1 57 

2 29 

3 

40 

47 

56 

1'07 

1 21 

1 38 

2 00 

2 34 

3 29 

4 

54 

1'04 

1'16 

1 33 

1 54 

2 24 

3 11 

4 38 

8 15 

5 

1'23 

1 41 

2 05 

2 41 

3 40 

5 40 

12  02 

1 

For  Latitude  47' 

3 30'. 

Oh. 

30" 

36" 

44" 

52" 

1'02" 

114" 

1'29" 

1'49" 

218" 

2 

35 

42 

50 

l'OO 

1 12 

1 26 

1 45 

2 01 

2 51 

3 

43 

51 

101 

1 13 

1 28 

1 47 

2 15 

2 56 

4 08 

4 

56 

1'09 

1 23 

1 40 

2 05 

2 40 

3 39 

5 37 

11  18 

5 

1'27 

1 46 

2 12 

2 52 

4 01 

6 30 

16  19 

For  Latitude  50°. 

Oh. 

33" 

40" 

48" 

57" 

1'08" 

1'21" 

1'39" 

2' 02" 

2'36" 

2 

38 

46 

55 

1'06 

1 18 

1 35 

1 57 

2 28 

3 19  1 

3 

47 

56 

1'06 

1 19 

1 36 

2 29 

2 31 

3 23 

5 02 

4 

102 

1'44 

1 29 

1 48 

2 16 

2 58 

4 18 

6 59 

19  47 

5 

1 30 

1 51 

2 19 

3 04 

4 22 

7 28 

2410 

For  Latitude  52 

§3 

Oh. 

36" 

44" 

52" 

102" 

114" 

1'29" 

1'49" 

218" 

3 05" 

2 

43 

50 

59 

1 11 

1 26 

1 42 

2 23 

2 49 

3 55 

3 

50 

l'OO 

I'll 

l 26 

1 45 

2 11 

2 51 

2 58 

6 22 

4 

1'05 

1 18 

1 35 

2 10 

2 28 

3 19 

4 53 

8 42 

5 

1 34 

1 56 

2 27 

3 16 

4 47 

8 52 

12 


ELEMENTS  OP  SURVEYING. 


[APP.  A. 


H 

►4 

I) 

E C L I 

NAT 

IONS 

'A 

< 

pH 

For  I 

LATITUDE  55°. 

B 

o 

H 

+ 

20° 

+ 15° 

+ 10° 

+ 5° 

0° 

— 5° 

— 10° 

— 15c 

— 

20° 

Oh. 

40' 

48" 

57" 

1'08" 

1'21" 

139" 

2'02" 

2'36" 

3 

33" 

2 

40 

55 

1'05 

1 18 

1 34 

1 56 

2 30 

3 15 

4 

47 

3 

55 

1 06 

1 19 

1 35 

1 58 

2 30 

3 21 

4 58 

9 

19 

4 

1 

'10 

1 23 

1 42 

2 06 

2 43 

3 44 

5 49 

12  41 

5 

1 

37 

2 01 

2 34 

3 28 

5 15 

10  18 

For  Latitude  57 

30'. 

Oh. 

44" 

52" 

1'02" 

1'14" 

1'29" 

1'49" 

2'18" 

3' 05' 

4 

’37" 

2 

50 

59 

1 11 

1 25 

1 43 

2 09 

2 47 

3 51 

6 

04 

3 

58 

1T0 

1 24 

1 42 

2 07 

2 43 

3 45 

5 50 

12 

47 

4 

1 

'll 

1 25 

1 43 

2 10 

2 50 

3 55 

6 14 

14  49 

5 

1 

41 

2 06 

2 42 

3 42 

5 46 

12  26 

Explanation  of  the  Table  of  Refractions.— The  table  is  cal- 
culated for  latitudes  between  30°  and  50  at  intervals  of  2|°,  that  being  as  near 
as  is  required. 

The  declination  ranges  from  0 to  20°  both  north  and  south,  the  + declina- 
tions being  north,  and  — south,  and  is  given  for  every  five  degrees,  that 
being  sufficiently  near  for  all  practical  purposes. 

The  hour  angle  in  the  first  column  indicates  the  distance  of  the  sun  from 
the  meridian  in  hours,  the  refraction  given  for  0 hours  b^ing  that  which 
affects  the  observed  declination  of  the  sun  when  on  the  meridian,  commonly 
known  as  meridional  refraction  ; the  refraction  for  the  hours  just  before  and 
after  noon  is  so  nearly  that  of  the  meridian,  that  it  may  be  called  and  allowed 
as  the  same. 

When  the  table  is  used,  it  must  be  borne  in  mind  that  when  the  declina- 
tion is  north  or  -f  in  the  table,  the  refraction  is  to  be  added  ; when  the  decli- 
nation is  south  or  — , the  refraction  must  be  subtracted. 

It  will  be  noticed  that  the  refraction  in  south  or  — declination  increases 
very  rapidly  as  the  sun  nears  the  horizon,  showing  that  observations  should 
not  be  taken  with  the  sun  when  south  of  the  equator,  less  than  one  hour 
from  the  horizon. 

The  calculation  of  the  declination  for  the  different  hours  of  the  day  should 
of  course  bo  made  and  noted  before  the  surveyor  commences  his  work,  that 
ho  may  lay  off  the  change  from  lirur  to  hour,  from  a table  prepared  before- 
hand. 


APP.  A.] 


THE  SOLAli  COMPASS. 


13 


Solar  Attachment  to  Transit. 


The  Solar  Attachment  is  essentially  the  solar  apparatus  of  Burt  placed 
upon  the  cross-bar  of  the  ordinary  transit,  the  polar  axis  only  being  directed 
above  instead  of  below,  as  in  the  solar  compass. 

A little  circular  disc  of  an  inch  and  a half  in  diameter,  and  having  a snort 
rouud  pivot  projecting  above  its  upper  surface,  is  first  securely  screwed  to 
the  telescope  axis. 

Upon  this  pivot  rests  the  enlarged  base  of  the  polar  axis,  which  is  also 
firmly  connected  with  the  disc  by  four  capstan  head  screws  passing  from  the 
under  side  of  the  disc  into  the  base  already  named. 

These  screws  serve  to  adjust  the  polar  axis. 

The  hour  circle  surrounding  the 
base  of  the  polar  axis  is  easily  mov- 
able about  it,  and  can  be  fastened 
at  any  point  desired  by  two  flat- 
head  screws  above.  It  is  divided 
to  five  minutes  of  time  ; is  figured 
from  I.  to  XII.,  and  is  read  by  a 
small  index  fixed  to  the  declination 
circle,  and  moving  with  it. 

A hollow  cone,  or  socket,  fitting 
closely  to  the  polar  axis  and  made 
to  move  snugly  upon  it,  or  clamped 
at  any  point  desired  by  a milled- 
head  screw  on  top,  furnishes,  by 
its  two  expanded  arms  below,  a 
firm  support  for  the  declination 
arc,  which  is  securely  fastened  to 
it  by  two  large  screws,  as  shown. 

The  declination  arc  is  of  about 
five  inches  radius,  is  divided  to 
quarter  degrees,  and  reads  by  its 
vernier  to  single  minutes  of  arc, 
the  divisions  of  both  vernier  and 
limb  being  in  the  same  plane. 

The  declination  arm  has  the 
usual  lenses  and  silver  plates  on 
the  two  opposite  blocks,  made  pre- 
cisely like  those  of  the  ordinary 
solar  compass,  but  its  vernier  ,is 
outside  the  block,  and  more  easily  read. 

The  declination  arm  has  also  a clamp  and  tangent  movement,  as  shown 
in  the  cut.  The  arc  of  the  declination  limb  is  turned  on  its  axis  and  one  or 
the  other  solar  lens  used,  as  the  sun  is  north  or  south  of  the  equator  ; the  cut 
above  shows  its  position  when  it  is  north. 

The  latitude  is  set  off  by  means  of  a large  vertical  limb  having  a radius  of 
two  and  a half  inches  ; the  arc  is  divided  to  twenty  minutes,  is  figured  from 
the  centre,  each  way,  up  to  80°,  and  is  read  by  its  vernier  to  single  minutes. 


Fig.  3, 

Showing  a Transit  with  Solar  Attachment. 


14  ELEMENTS  OF  SURVEYING.  [APP.  A. 

It  has  also  a clamp-screw  inserted  near  its  centre,  by  which  it  can  be  set 
fast  to  the  telescope  axis  in  any  desired  position. 

The  vernier  of  the  vertical  limb  is  made  movable  by  the  tangent-screw 
attached,  so  that  its  zero  and  that  of  the  limb  are  readily  made  to  coincide 
when,  in  adjusting  the  limb  to  the  level  of  the  telescope,  the  arc  is  clamped 
to  the  axis. 

The  usual  tangent  movement  to  the  telescope  axis  serves,  of  course,  to 
bring  the  vertical  limb  to  the  proper  elevation,  as  hereafter  described. 

A level  on  the  under  side  of  the  telescope,  with  ground  vial  and  scale,  is 
indispensable  in  the  use  of  the  Solar  attachment. 

To  Find  the  Latitude.— First  level  the  instrument  very  carefully 
using  the  level  of  the  telescope  until  the  bubble  will  remain  in  the  centre 
during  a complete  revolution  of  the  instrument,  the  tangent  movement  of  the 
telescope  being  used  in  connection  with  the  leveling  screws  of  the  parallel 
plates,  and  the  axis  of  the  telescope  firmly  clamped. 

Next  clamp  the  vertical  arc  so  that  its  zero  and  that  of  its  vernier  coincide 
as  near  as  may  be,  and  then  bring  them  into  exact  line  by  the  tangent-screw 
of  the  vernier. 

Then,  having  the  declination  of  the  sun  for  12  o’clock  of  the  given  day  as 
affected  by  the  meridional  refraction  carefully  set  off  upon  the  declination  arc, 
note  also  the  equation  of  time  and  fifteen  or  twenty  minutes  before  noon,  the 
telescope  being  directed  to  the  north,  and  the  object  end  lowered  until,  by 
moving  the  instrument  upon  its  spindle  and  the  declination  arc  from  side  to 
side,  the  sun’s  image  is  brought  nearly  into  position  between  the  equatorial 
lines.  Now  bring  the  declination  arc  directly  in  line  with  the  telescope, 
clamp  the  axis  firmly,  and  with  the  tangent  screw  bring  the  image  precisely 
between  the  lines  and  keep  it  there  with  the  tangent  screw,  raising  it  as  long 
as  it  runs  below  the  lower  equatorial  line,  or,  in  other  words,  as  long  as  the 
sun  continues  to  rise  in  the  heavens. 

When  the  sun  reaches  the  meridian  the  image  will  remain  stationary  for 
an  instant  and  then  begin  to  rise  on  the  plate. 

The  moment  the  image  ceases  to  run  below  is  of  course  apparent  noon, 
when  the  index  of  the  hour  arc  should  indicate  XII.,  and  the  latitude  be  deter- 
mined by  the  reading  of  the  vertical  arc. 

It  must  be  remembered,  however,  that  the  angle  through  which  the  polar 
axis  has  moved  in  the  operation  just  described  is  measured  from  the  zenith 
instead  of  the  horizon  as  in  the  ordinary  solar,  so  that  the  angle  read  on  the 
vertical  limb  is  the  complement  of  the  latitude. 

The  latitude  itself  is  readily  found  by  subtracting  this  angle  from  90°  ; 
thus  at  Troy,  the  reading  of  the  limb  being  found  as  above  directed  to  be 
47°  10',  the  latitude  will  be  90° -47°  16' ^42°  44'. 

It  will  be  noticed  that  with  this  apparatus  the  latitude  of  any  place  can 
bo  most  easily  ascertained  without  any  index  error,  as  in  the  usual  solar 
compass. 

To  run  lines  with  the  Solar  Attachment — Having  set  off 
the  complement  of  the  latitude  of  the  place  on  the  vortical  arc,  and  the  decli- 
nation for  the  given  day  and  hour  as  in  the  solar,  the  instrument  being  also 
carefully  leveled  by  the  telescope  bubble,  sot  the  horizontal  limb  at  zero  and 


APP.  A.] 


THE  SOLAR  COMPASS. 


15 


clamp  the  plates  together,  loosen  the  lower  clamp  so  that  the  transit  moves 
easily  upon  its  lower  socket,  set  the  instrument  approximately  north  and 
south,  the  object  end  of  the  telescope  pointing  to  the  north,  turn  the  proper 
solar  lens  to  the  sun,  and  with  one  hand  on  the  plates  and  the  other  on  the 
revolving  arm,  move  them  from  side  to  side  until  the  sun’s  image  is  brought 
between  the  equatorial  lines  on  the  silver  plate. 

The  lower  clamp  of  the  instrument  should  now  be  fastened,  and  any  fur- 
ther lateral  movement  be  made  by  the  tangent  screw  of  the  tripod.  The 
necessary  allowance  being  mado  for  refraction,  the  telescope  will  be  in  the 
true  meridian,  and  being  unclamped,  may  be  used  like  the  sights  of  the  ordi- 
nary solar  compass,  but  with  far  greater  accuracy  and  satisfaction  in  estab- 
lishing meridian  lines.  Of  course  when  the  upper  or  vernier  plate  is  un- 
damped from  the  limb,  any  angle  read  by  the  verniers  is  an  angle  from 
the  meridian,  and  thus  parallels  of  latitude  or  any  other  angles  from  the  true 
meridian  may  be  established  as  with  the  solar  compass. 

The  bearing  of  the  needle,  when  the  telescope  is  on  the  meridian,  will  also 
give  the  variation  of  the  needle  at  the  point  of  observation. 

If  the  instrument  has  a movable  compass  circle,  the  variation  of  the 
needle  can  be  set  off  to  single  minutes,  the  needle  kept  at  zero,  or  “ with  the 
sun,”  and  thus  lines  be  run  by  the  needle  alone  when  the  sun  is  obscured. 


APPENDIX  B. 


THE  SEXTANT* 

(By  Prop.  J.  K.  Rees,  Columbia  College.) 

This  instrument  is  especially  useful  to  the  scientific  explorer  on  account 
of  its  portability  and  simplicity  of  manipulation.  It  requires  no  fixed  sup- 
port, and  furnishes  data  with  the  least  expenditure  of  the  time  of  the  ob- 
server. The  accuracy  of  fixed  instruments  is  not  to  be  expected  from  it, 
since  it  is  held  in  the  hand  and  is  of  small  dimensions. 

The  Principle  of  the  Instrument.— The  optical  principle  upon 
which  the  sextant  is  made 
is  : — If  a ray  of  light  suffers 
two  successive  reflections  in 
the  same  plane  by  two  plane 
mirrors,  the  angle  between 
the  first  and  last  directions 
of  the  ray  is  twice  the  angle 
of  the  mirrors. 

Let  I and  H be  two 
plane  mirrors  perpendicular 
to  the  plane  of  the  paper, — 
which  is  taken  as  the  plane 
of  reflection. 

A ray  of  light  from  A is 
reflected,  first  from  the  mir- 
ror 1 in  the  direction  GO, 
then  by  the  mirror  II  along 
OT.  The  angle  between 
Hie  first  and  last  direction 
of  the  ray  after  these  two 
reflections  is  A TO. 

Draw  ON  and  OM  nor- 
mal to  the  mirrors  I and  7/ 
respectively.  Then  NKO 
equals  the  angle  of  the  mirrors. 

From  the  law  of  the  reflection  of  light  it  is  known  that  the  angle 
A ON  or  i — anglo  NGO  ; 
and  also,  COM  or  i!  - MOT. 

* For  u full  description  of  this  Instrument,  sec  Chauvenefs  Spherical  and  Practical 
Astronomy,  published  by  Llppincott  in  1808. 


JL 


APP.  B.] 


THE  SEXTANT. 


17 


ce  angle  A10  — 2i  — 2i'  ; 

angle  NKO  — i — i'  ; 

ATO  = 2 NKO. 


Q.  E.  D. 


Suppose  now  that  the  glass  His  unsilvered  on  the  upper  half ; then  a ray 
of  light  coming  from  B will  pass  through  this  unsilvered  portion  to  T,  and 
the  angle  ATO  will  measure  the  angular  distance  of  A from  B as  seen  at  T. 

To  apply  this  principle  the  mirror  I,  revolving  about  a pivot  at  G,  has  at- 
tached to  it  an  arm  or  bar  ID,  which,  as  the  mirror  is  turned,  moves  over  a 
graduated  arc  ER.  The  mirror  H is  fixed  in  position.  There  will  be  one 
position  of  the  index  arm  where  the  two  mirrors  will  be  parallel.  Then  since 
the  angle  bet  wean  the  first  and  last  directions  of  the  ray  of  light  which  is 
reflected  by  both  mirrors  is  zero,  or  the  two  directions  are  parallel,  the  indi- 
cated point  of  the  graduated  arc  is  marked  zero.  The  graduations  are  then 
continued  to  the  left,  calling  each  degree  two  degrees,  in  order  to  read  off  at 
once  the  required  angle. 

The  best  form  of  the  common  sextant  is  seen  in  the  accompanying  cut, 
furnished  by  Messrs.  Stackpole  & Brother,  N.  Y.  : 


The  frame  is  of  brass  » 
constructed  so  as  to  com- 
bine strength  with  light- 
ness ; the  graduated  arc, 
inlaid  in  the  brass,  is  usu  Q 
ally  of  silver.  The  divi- 
sions of  the  arc  are  ordi- 
narily 10'  each,  which  are 
subdivided  by  the  vernier 
to  10".  The  handle  A by 
which  it  is  held  in  the  hand 
is  of  wood. 

The  mirrors  I and  H 
are  of  plate  glass  silvered. 

The  upper  half  of  the  glass 
II  is  left  unsilvered  in  or- 
der that  the  direct  rays 
from  a distant  object  may  not  be  interrupted.  To  give  greater  distinctness 
to  the  images  a small  telescope  T is  placed  in  the  line  of  sight  OT.  The 
telescope  is  supported  in  a ring  D,  which  can  be  moved  in  a direction  at 
right  angles  to  the  plane  of  the  sextant.  Thus  the  axis  of  the  telescope  can 
be  directed  either  towards  the  silvered  or  the  unsilvered  part  of  the  mirror. 
This  motion  changes  the  plane  of  reflection,  which,  however,  remains 
always  parallel  to  the  plane  of  the  sextant ; the  use  of  the  motion  being 
merely  to  regulate  the  relative  brightness  of  the  direct  and  reflected  images. 
The  vernier  is  read  with  the  aid  of  a glass  G which  is  attached  to  the  ii\dex 
bar.  The  central  mirror  I,  or  index  glass,  is  fastened  in  a brass  frame  which 
is  firmly  attached  to  the  index  bar  by  three  screws.  This  glass  is  generally 
set  by  the  maker  so  as  to  be  perpendicular  to  the  plane  of  the  sextant.  There 
are  no  adjusting  screws  usually  connected  with  it.  The  fixed  mirror  II,  or 
horizon  glass  (so  called  because  through  it  the  horizon  is  observed  in  taking 


ELEMENTS  OF  SURVEYING. 


18 


[app.  b. 


altitudes),  is  specially  provided  with  screws  by  which  its  position  with  respect 
to  the  sextant  plane  may  be  rectified. 

At  P and  Q are  colored  glasses  of  different  shades,  which  may  he  used 
separately  or  in  combination,  to  defend  the  eye  from  the  intense  light  of  the 
sun. 

Common  Adjustments  of  Ordinary  Sextant. 

1.  The  Index  Glass  must  he  perpendicular  to  the  plane  of  the  sextant. 

2.  The  Horizon  Glass  must  also  he  perpendicular  to  the  plane  of  the  sextant. 

3.  The  central  sight  line  of  the  telescope  must  he  parallel  to  the  plane  of  the 
sextant. 

4-  The  true  zero  of  the  arc  must  he  found. 

1.  Adjustment  of  the  Index  Glass.— Bring  the  vernier  to 
about  the  middle  of  the  graduated  arc  ; then,  placing  the  eye  a little  above 
the  plane  of  the  sextant  and  near  the  index  glass,  examine  the  direct  and  re- 
flected images  of  the  graduated  arc.  If  the  one  appears  to  run  into  the  other 
the  index  glass  is  perpendicular  to  the  plane  of  the  sextant,  and  the  adjust- 
ment is  complete. 

If  the  reflected  image  appears  too  high  or  too  low  the  glass  leans  forward 
or  backward.  The  glass  may  then  be  adjusted  to  perpendicularity  by  placing 
a piece  of  paper  under  one  edge  of  the  plate  by  which  the  glass  is  held  to 
the  index  arm,  first  loosening  the  screws  ; or  the  glass  may  be  taken  out  of 
the  frame,  and  the  supports  against  which  the  glass  leans  may  be  filed  so  as 
to  bring  the  glass,  when  set  back,  perpendicularly  to  the  plane  of  the  sextant. 

2.  Adjustment  of  the  Horizon  Glass.— This  glass  must  also 
be  perpendicular  to  the  plane  of  the  sextant.  The  index  glass  having  been 
adjusted  to  perpendicularity,  if  it  is  found  that  in  any  one  position  the  hori- 
zon glass  is  parallel  to  the  index  glass,  then  the  horizon  glass  is  perpendicular 
to  the  plane  of  the  sextant. 

In  order  to  test  this  parallelism,  put  in  the  telescope  and  direct  it  to  a star 
or  any  distant,  well-defined  terrestrial  object.  Move  the  index  bar  until  the 
direct  and  reflected  images  are  in  the  field  of  view,  then  clamp  the  vernier, 
and  by  moving  the  tangent  screw  cause  one  image  to  pass  the  other  ; if  they 
pass  exactly  one  over  the  other  the  adjustment  is  complete  ; if  they  pass  one 
at  the  side  of  the  other,  the  horizon  glass  must  be  adjusted.  There  are  ad- 
justing screws  attached  to  the  glass  whereby  it  can  be  inclined  to  or  from  the 
sextant  plane,  and  also  turned  around  an  axis  perpendicular  to  the  sextant 
plane.  By  means  of  the  first  set  of  screws  the  adjustment  for  perpendicu- 
larity can  be  made,  and  by  means  of  the  second  set  the  position  of  the  zero  of 
the  limb  can  be  altered  to  a small  extent. 

3.  Adjustment  Of  the  Telescope.— The  sight  line  of  the  tele- 
scope is  the  line  from  the  centre  of  the  field  of  view  through  the  centre  of  the 
object  glass.  This  line  must  be  parallel  to  the  plane  of  the  sextant.  In 
order  to  test  for  this,  choose  two  distant  objects  like  the  sun  and  moon,  90°  to 
120  apart ; direct  the  telescope  to  one  of  these  objects,  holding  tho  plane  of 
the  sextant  so  as  to  pass  through  both  ; then  moving  the  index  bar,  bring 
the  second  object  into  the  field  of  viow  ; clamp  the  vernier,  turn  the  tangent 


APP.  B.] 


THE  SEXTANT. 


19 


screw  until  the  two  objects  are  tangent  to  each  other  on  the  thread  of 
the  telescope  nearest  to  the  instrument.  Then  by  moving  the  instrument, 
cause  the  objects  to  come  on  the  thread  farthest  from  the  instrument.  If  the 
tangency  is  still  perfect,  the  adjustment  is  complete.  If  the  objects  separate 
upon  the  thread  farthest  from  the  instrument,  then  the  object  end  of  the 
telescope  droops  towards  the  plane  of  the  instrument  ; if  the  images  overlap, 
then  the  telescope  inclines  upward  from  the  plane  of  the  instrument.  The 
adjustment  is  made  by  meaus  of  the  screws  that  work  into  the  collar  which 
carries  the  telescope. 

4.  Index  Correction. — When  the  two  glasses  are  parallel  the  zero 
of  the  graduated  limb  should  coincide  with  the  zero  of  the  vernier.  This 
adjustment  must  be  very  carefully  looked  after  before  taking  any  observa- 
tions, because  it  is  an  adjustment  that  is  liable  to  change.  Rather  than  make 
this  adjustment  accurately  every  time  an  observation  is  made  it  is  prefer- 
able to  determine  the  place  of  the  true  zero  of  the  graduated  arc  and  allow 
for  the  correction.  This  correction  is  simply  the  distance  between  the  gradu- 
ated zero  of  the  instrument  and  the  reading  of  the  vernier  when  the  two 
mirrors  are  parallel.  This  correction  is  minus  when  it  is  on  the  graduated 
arc  towards  the  increasing  numbers  of  the  graduation,  and  plus  when  on  the 
opposite  side. 

The  index  correction  may  be  determined  by  observations  upon  a star,  or  a 
distant,  well-defined,  terrestrial  object,  or  upon  the  sun. 

First,  by  a star  : Direct  the  telescope  to  a star  of  the  third  or  fourth  mag- 
nitude ; move  the  index  bar  until  the  reflected  image  of  this  star  comes  into 
the  field  of  view ; then  clamp  the  vernier  ; turn  the  tangent  screw  until  the 
direct  and  reflected  images  of  the  star  are  exactly  in  coincidence  ; take  the 
reading  of  the  vernier;  apply  the  proper  sign,  and  the  arc  reading  is  the 
index  correction. 

Second,  by  a distant,  well-defined  terrestrial  object,  or  the  reflection  of 
sunlight  from — for  example — the  bulb  of  a thermometer  or  a drop  of  water. 
This  can  be  observed  in  the  same  way  as  a star,  although  not  giving  as  accu- 
rate results. 

Third,  by  the  sun  : Turn  on  the  colored  glasses  until  the  light  from  the 
sun  is  diminished  sufficiently  to  suit  the  eye  ; bring  the  direct  and  reflected 
images  of  the  sun  into  the  field  of  view  ; clamp  the  vernier ; turn  the  tangent 
screw  until  one  image  of  the  sun  is  tangent  to  the  other ; take  the  reading ; 
turn  the  tangent  screw  until  the  contact  is  broken  ; bring  the  images  back  to 
tangency  ; take  the  reading  again  ; in  this  way  make  five  readings  ; then  turn 
the  tangent  screw  until  the  images  change  places  and  tangency  is  made  on 
the  other  side  ; take  the  same  number  of  readings  here. 

In  order  to  read  always  from  the  same  end  of  the  vernier,  call  the  zero  of 
the  vernier  860°,  and  read  the  vernier  accordingly.  Take  the  mean  of  the 
readings  in  the  first  and  second  cases,  add  them  together,  and  divide  by  two. 
Subtract  the  result  from  360°,  and  the  difference  will  be  the  index  correction. 

In  order  to  avoid  the  effect  of  refraction  it  is  best  to  measure  the  horizontal 
diameter  of  the  sun.  To  check  the  observations  compute  the  diameter  of  the 
sun  from  data  given  in  the  Nautical  Almanac,  for  the  day  of  the  observation, 
and  compare  it  with  the  diameter  of  the  sun  as  obtained  from  the  obser- 
vations. 


20 


ELEMENTS  OF  SURVEYING. 


[API*.  B. 


Note— Let  R be  the  true  reading  of  the  vernier  when  the  mirrors  are 
parallel;  let  8 be  the  diameter  of  the  sun  ; let  r be  the  reading  of  the  vernier 
when  contact  is  made  on  the  left  of  the  zero  of  the  instrument,  and  r'  the 
reading  when  the  contact  is  made  on  the  right. 


Then, 


Hence, 


and 


r = 
r'  = 


R — 


8 = 


R + 8 
R - 8. 
r + r' 
~2~’ 
r — r' 

2 


Observed  sun  for  iudex  correction. 
ON  ARC. 

360°  34'  10" 

10" 

5 

10 

0 


June  1st,  1883—10  a.  m. 

off  ARC. 

359  30  50 
45 
45 
55 
50 


360  34 

07  = r 

means  359 

30 

49  = 

359  '30 

49 

360 

34 

07 

2)720  4 

56 

2)  1 

3 

18 

360  2 

28 

Observed  diam.  sun, 

31 

39. 

Index  correction. 

Calculated  diam.  sun, 

31 

36.7 

-2'  28" 

Difference,  . . . 

2.3 

To  Measure  the  Angular  Distance  between  two  objects  with  the 

Sextant. 

Turn  the  eye-piece  until  two  of  the  reticule  threads  of  the  telescope  are 
parallel  to  the  plane  of  the  instrument ; then  direct  the  telescope  to  the  fainter 
of  the  two  objects  ; move  the  plane  of  the  instrument  until  it  passes  through 
the  two  objects  ; revolve  the  index  bar  until  the  reflected  image  of  the  second 
object  appears  in  the  field  of  view  ; clamp  the  vernier  ; with  the  tangent 
screw  make  the  two  images  coincide.  The  reading  of  the  vernier  with  the 
index  correction  applied  will  be  the  angular  distance  of  the  two  objects.  The 
index  glass  must  always  be  on  the  side  towards  the  second  object.  The  coin- 
cidence must  be  made  at  the  middle  point  of  the  field  of  view.  Care  must 
bo  taken  to  have  the  relative  brightness  of  the  two  objects  about  the  same. 

Altitudes  of  objects  may  be  obtained  in  a similar  way,  holding  the  plane 
of  the  sextant  vertical. 


APPENDIX  C. 


INSTRUCTIONS  TO  DEPUTY  U.  S.  MINERAL 
SURVEYORS. 


Surveyor  General’s  Office, 

Denver,  Colorado,  June  1,  1880. 

Sir  : — The  following  instructions  have  been  prepared  for  your  government 
in  making  mineral  surveys  : 

1.  You  can  make  no  official  survey  without  an  order  from  this  office. 

2.  To  procure  an  order  for  survey,  the  application  should  be  made  in 
writing,  accompanied  with  a copy  of  the  certificate  of  location  of  the  claim  to 
be  surveyed,  duly  certified  by  the  proper  recorder,  and  inclosing  a deposit  of 
$25  for  office  work.  Be  careful  to  state  plainly  the  name  of  the  person  or 
persons  who  desire  the  survey. 

8.  An  order  will  not  be  issued  unless  in  the  judgment  of  this  office  the 
certificate  of  location  is  in  accordance  with  law  and  regulations. 

4.  For  the  accommodation  of  claimants  residing  remote  from  a United 
States  Depository,  this  office  will  see  that  the  deposit  is  made  when  the 
proper  funds  are  received.  Send  Post  Office  order,  draft  or  check,  on  Denver, 
as  other  drafts  cannot  be  used  without  discount. 

5.  Should  the  original  certificate  of  location  be  found  defective,  and  the 
description  does  not  cover  the  ground  claimed,  an  additional  or  amended  cer- 
tificate may  be  filed  as  provided  by  section  13  of  the  laws  of  Colorado,  ap- 
proved February  13th,  1874,  and  a certified  copy  of  the  amended  certificate 
must  be  sent  to  this  office,  and  your  survey  dated  subsequent  to  the  filing  of 
the  same  in  the  proper  recorder’s  office. 

6.  Certificates  of  location  are  frequently  sent,  which  any  well-informed 
deputy  should  KNOW  are  too  indefinite  ; examine  them  carefully  before  send- 
ing, and  it  will  save  yourselves  and  this  office  much  trouble,  and  remember 
that  your  survey  must  agree  with  certificate  of  location. 

7.  “ If  the  records  of  locations  prior  to  the  passage  of  the  mining 
act  of  May  10th,  1872,  are  not  sufficiently  definite  and  certain  to  enable  you 
to  make  a correct  survey  therefrom,  you  should,  after  reasonable  notice  in 
writing,  to  be  served  personally,  or  through  the  United  States  mail,  on  the 
applicant  for  survey  and  adjoining  claimants  (whose  residence  or  Post  Office 
address  you  may  know  or  can  ascertain  by  the  exercise  of  reasonable  dili- 
gence), take  the  testimony  of  neighboring  claimants  and  other  persons  who 
are  familiar  with  the  boundaries  thereof  as  originally  located  and  asserted  by 
the  locators  of  the  claim,  and  after  having  ascertained  by  such  testimony  the 


22 


ELEMENTS  OF  SURVEYING. 


[APP.  C. 


boundaries  as  originally  established,  you  should  make  a survey  in  accordance 
therewith,  and  transmit  full  and  correct  returns  of  survey,  accompanied  by 
the  copy  of  the  record  of  location,  the  testimony,  and  a copy  of  the  notice 
served  on  the  claimant  and  adjoining  proprietors,  certifying  thereon  when,  in 
what  manner,  and  on  whom,  service  was  made.” 

8.  The  act  of  Congress  of  May  10th,  1872,  expressly  provides  that  “ the 
location  must  be  distinctly  marked  on  the  ground,  so  that  its  boundaries  can 
be  readily  traced,”  and  “ that  all  records  of  mining  claims  hereafter  made 
shall  contain  the  name  or  names  of  the  locators,  the  date  of  location,  and 
such  a description  of  the  claim  or  claims,  located  by  reference  to  some  natural 
object  or  permanent  monument,  as  will  identify  the  claim.” 

9.  “ These  provisions  of  the  law  must  be  strictly  complied  with  in  each 
case  to  entitle  a claimant  to  a survey  and  patent,  and  therefore  should  a claim- 
ant under  a location  made  subsequent  to  the  passage  of  the  mining  act  of  May 
10th,  1872,  who  has  not  complied  with  said  requirements  in  regard  to  mark- 
ing the  location  upon  the  ground, and  recording  the  same,  apply  fora  survey,” 
I “will  decline  to  order  it.” 

10.  “ The  only  relief  for  a party  under  such  circumstances  will  be  to  make 
a new  location  in  conformity  to  law  and  regulations,  as  no  case  will  be  ap- 
proved by  this  office  unless  these  and  all  other  provisions  of  law  are  substan- 
tially complied  with.” 

11.  Corners  must  be  established  in  as  permanent  a manner  as  possible, 
and  should  consist  of  rock  in  place,  or  tree  if  they  are  found  at  the  exact  point ; 
otherwise  a STONE,  not  less  than  two  feet  long,  set  one  foot  in  the  ground  ; 
or  POSTS,  not  less  than  four  inches  in  diameter , planted  two  feet  in  the  ground 
and  protruding  not  less  than  two  and  not  more  than  four  feet  above  ground  ; 
both  stones  and  posts  must  be  protected  with  a mound  of  stone  or  earth  in 
addition  to  the  planting. 

12.  Each  corner  to  be  marked  No.  1,  2,  etc.,  also  the  number  of  the  sur- 
vey, as  you  proceed  with  the  work,  givingin  field  notes,  bearings  and  distances 
from  each  corner  to  rocks  or  trees,  if  any  such  are  at  convenient  distance, 
marking  same  with  number  of  corner  and  number  of  survey,  and  describe 
marks.  Wooden  posts  and  trees  must  be  marked  with  a scribe,  and  rocks 
with  a chisel.  Give,  from  at  least  two  corners  of  the  survey,  two  or  more 
bearings  to  well  known  points,  such  as  mountain  peaks,  confluence  of  streams, 
etc.,  both  in  field  notes  and  on  plats.  If  any  portion  of  this  section  cannot 
be  complied  with,  so  state  it  in  field  notes. 

13.  Note  all  objects  crossed  by  your  line  of  survey,  such  as  previous  sur- 
veys, lodes,  ditches,  roads,  ravines,  etc., etc.,  and  show  them  upon  plat.  “ If 
in  running  the  exterior  boundaries  of  a claim  it  is  found  that  two  surveys 
conflict,  the  plats  and  field  notes  should  show  the  extent  of  the  conflict,  giv- 
ing the  area  embraced  in  both  surveys,  and  also  the  courses  and  distances 
from  the  established  corners  at  which  the  exterior  boundaries  of  the  respec- 
tive surveys  intersect  each  other.  In  notes  give  area  as  follows : 


Total  area, 5.94  acres. 

Less  area  in  conflict  with  surveys  Nos.  13  and  17,  . . 2.00  acres. 

Leaving  net  area, 3.94  acres. 

On  plat  give  net  area  only. 


APP.  C.]  INSTRUCTIONS  TO  U.  S.  MINERAL  SURVEYORS. 


23 


14.  Connect  corner  No.  1 of  your  survey  with  some  corner  of  the  public 
survey  if  the  claim  is  located  within  two  miles  of  such  public  survey.  From 
corner  No.  1 beginning  you  will  proceed  with  the  survey  of  the  claim,  giving 
distances  in  feet  and  true  courses,  as  you  proceed  establishing  a corner  at  each 
angle  of  the  survey.  If  an  official  survey  has  been  made  in  the  same  vicinity, 
run  a connecting  line  to  some  corner  of  the  same.  Surveys  must  be  made  to 
close  exactly.  Mention  particularly  all  adjoining  claimants. 

15.  In  referring  to  other  surveys  give  in  your  field  notes  the  name  and 
number  of  each  survey,  and  name  of  claimant  or  owner.  When  referring  to 
a survey  more  than  once  it  is  unnecessary  to  repeat  more  than  the  number. 

16.  See  Section  2320,  Revised  Statutes  of  the  United  States,  in  regard  to 
width  of  lode  claims  “ on  each  side  of  the  middle  of  the  vein  at  the  surface.” 
When  the  locator  does  not  determine,  by  exploration,  where  the  middle  of  the 
vein  at  the  surface  is,  his  discovery  shaft  must  be  assumed  to  mark  such 
point.” 

17.  You  will  give  the  quarter  section,  township  and  range  in  which  the 
claim  is  situated,  in  notes  and  on  plat,  showing  section  lines  on  your  plat  in 
black,  and  quarter  section  lines  in  red. 

18.  In  districts  where  there  are  no  public  surveys  within  two  miles  of 
your  survey,  and  no  locating  monument  previously  established  within  one 
hundred  chains  of  such  survey,  you  will  establish  a LOCATING  MONU- 
MENT at  some  well-known  point,  a rock  in  place  being  preferable  ; but  if 
such  cannot  be  found,  then  erect  a large  substantial  mound  of  rocks,  describing 
the  same  fully  in  your  notes,  and  hereafter  chisel  upon  it  a name,  using  this 
as  a starting  point  for  your  surveys,  and  when  the  public  surveys  reach  the 
locality,  run  a connecting  line  from  the  monument  to  a corner  in  said  surveys, 
thus  connecting  all  claims  surveyed  from  the  monument. 

19.  Note  all  improvements  upon  the  claim,  such  as  shafts,  drifts,  adits, 
cuts,  buildings,  etc. , giving  the  extent  of  same.  Show  improvements  on  plat 
and  locate  them  in  field  notes,  by  course  and  distance,  in  a direct  line  from 
some  corner  of  the  survey. 

20.  After  describing  fully  the  improvements  (stating  if  the  excavations 
are  in  dirt  or  rock)  placed  on  the  claim  by  the  claimant  or  his  grantors,  say 
that  “ the  value  of  the  said  improvements,  together  with  the  labor  expended 
on  the  said  claim  by  the  claimant  (or  his  grantors,  as  the  case  may  be)  is  not 
less  than  five  hundred  dollars.”  If  $500  in  improvements  has  not  been  ex 
pended  at  time  of  survey  the  work  maybe  performed  and  certificate  of  Sur- 
veyor-General filed  at  the  land  office  during  the  60  days  of  advertising.  To 
obtain  this  certificate  the  deputy  must  send  his  affidavit  stating  that  the  sum 
of  $500  has  been  expended,  describing  improvements  fully.  In  estimating 
the  value  of  improvements  only  actual  mining  improvements  should  be  con- 
sidered. 

21.  “ From  an  examination  of  the  returns  of  surveys  of  mining  claims,  I 
am  satisfied  that  in  many  instances  the  deputy  surveyors  certify  to  the  value 
of  improvements  without  ascertaining  whether  such  improvements  are  made 
by  the  claimant  or  his  grantors,  or  other  persons.” 

22.  “ No  improvements  should  be  included  in  the  estimate  unless  they 
have  been  made  by  the  applicant  for  survey,  or  by  those  from  whom  he  de- 
rives his  title,  and  so  stated  in  your  notes.” 

23.  “ The  value  of  improvements  made  upon  other  locations,  or  by  other 


ELEMENTS  OF  SURVEYING. 


H 


[app.  c. 


persons,  should  not  be  taken  into  consideration,  but  excluded  by  deputies  in 
tlieirestimate  of  improvements  upon  the  claim.” 

24.  Following  description  of  improvements  made  by  the  claimant,  locate 
the  improvements  made  by  other  persons,  in  the  manner  described  in  Sec.  19. 

25.  State  all  facts  coming  to  your  knowledge  in  regard  to  adjoining 
conflicting  claimants,  whether  their  claims  are  surveyed  or  not. 

26.  Your  field  notes  should  be  complete  in  themselves,  leaving  nothing 
to  be  explained  by  letter ; and  as  they  are  bound  in  book  form  after  approval 
you  should  leave  margin  for  binding.  Fold  notes  so  as  to  leave  as  little 
blank  paper  as  possible,  and  stitch  them  together. 

27.  If  you  know  your  survey  does  not  agree  with  that  of  other  deputies 
you  should  communicate  with  such  deputy  before  sending  your  survey  to 
this  office,  and  try  to  reconcile  all  discrepancies  ; if  it  cannot  be  adjusted,  re- 
port to  this  office,  and  a joint  survey  will  be  ordered  upon  knowing  that  this 
section  has  been  complied  with. 

28.  When  a joint  survey  is  ordered,  the  deputy  who  discovers  the  sup- 
posed error  will  be  directed  to  call  upon  the  deputy  supposed  to  be  in  error, 
and  see  if  the  discrepancy  cannot  be  reconciled.  If  they  cannot  agree,  then 
they  shall  make  a joint  survey  within  ten  days  after  the  date  of  the  order. 
When  said  survey  is  completed  they  shall  make  a joint  affidavit  to  the  field 
notes,  and  forward  them  to  this  office.  The  deputy  found  to  be  in  error  shall 
pay  all  expenses,  including  $10  per  day  to  the  deputy  whose  work  is  found 
to  be  correct.  If  both  deputies  are  found  to  be  in  such  error  as  to  require 
amended  field  notes  of  their  former  survey,  then  each  deputy  shall  pay  one- 
half  the  expenses  of  the  joint  survey.  Any  deputy  refusing  or  neglecting  to 
appear  for  the  joint  survey,  within  the  ten  days  named,  or  who  shall  refuse 
or  neglect  to  pay  the  expenses  as  above  indicated,  will  be  suspended  ; and  if 
the  refusal  or  neglect  shall  extend  to  twenty  days,  his  commission  as  deputy 
will  be  revoked. 

29.  In  preparing  plats  make  the  top  North,  and  color  only  the  ground  not 
in  conflict.  Mark  upon  the  plat  all  corners  (thus,  “ Cor.  No.  1,”  etc.),  courses 
to  mountain  peaks,  courses  and  distances  to  all  connecting  lines  and  upon  the 
boundaries  of  the  survey.  Plat  survey  upon  as  large  a scale  as  paper  will 
admit  of,  if  practicable  not  less  than  200  feet  to  the  inch.  Be  careful  that 
your  plats  and  notes  agree. 

30.  An  applicant  has  the  right  to  abandon,  from  his  application  for  patent, 
any  part  or  portion  of  the  premises  embraced  in  the  survey,  but  in  case  he 
does  abandon  any  portion  of  the  premises  embraced  by  his  application  and 
survey,  it  will  be  necessary  that  an  amended  survey  be  made  vpon  applica- 
tion of  the  claimant,  and  as  such  amended  survey  involves  as  much  office  work 
as  the  original  survey,  the  usual  deposit  should  accompany  the  claimant’s 
application  for  the  amended  survey. 

31.  After  this  date  surveys  will  bo  approved  in  the  order  in  which  they 
are  received,  deputies  being  required  to  act  promptly  in  all  official  matters, 
that  the  approval  of  a survey  may  not  bo  unnecessarily  delayed.  In  case  of 
unreasonable  delay  on  the  part  of  the  deputy  to  correct  errors,  the  survey  will 
be  stricken  from  the  files,  and  the  applicant  notified. 

32.  In  order  that  all  returns  may  be  made  uniform,  blank  plats  and  field 
note  paper  will  bo  furnished  from  this  office  for  your  use,  which  blanks  you 
will  (ill  up  as  the  cuse  may  require.  You  will  bo  careful  that  the  names  of 


APP.  C.]  INSTRUCTIONS  TO  U.  S.  MINERAL  SURVEYORS. 


25 


claimants  and  number  of  survey  agree  with  the  order  for  survey  and  certifi- 
cate  of  location.  You  will  make  one  copy  of  the  plat  and  notes  and  transmit 
the  same  to  this  office,  prepaying  the  postage  or  express,  in  full ; otherwise 
they  will  not  be  received.  Do  not  fold  plats  for  transmission,  but  roll  them. 
As  your  plat  and  notes  come  under  the  head  of  -‘written  matter,”  they 
require  letter  postage. 

33.  Unless  plats  and  notes  are  prepared  in  a neat  and  workmanlike  man 
ner  they  will  not  be  accepted.  Plats  and  notes  are  often  found  to  be  incor- 
rect from  negligence,  carelessness  and  ignorance.  Many  deputies  appear  to 
depend  upon  this  office  to  detect  and  correct  these  errors,  as  it  saves  them  the 
trouble.  Hereafter  such  surveys  will  be  returned  with  the  simple  statement 
that  “ they  are  incorrect.” 

34.  A solar  transit  must  be  used  in  all  official  surveys  being  guided  by 
the  solar  apparatus,  and  not  the  needle,  unless  the  courses  are  deflected  from 
a meridian  astronomically  established.  On  account  of  local  attractions  a 
needle  instrument  will  not  be  accepted  as  reliable.  State  in  field  notes  the 
kind  of  instrument  used,  and  the  manner  in  which  the  courses  are  taken. 

35.  “ You  are  informed  that  the  employment  of  Deputy  Mineral  Survey- 
ors as  attorneys  in  mineral  claims,  directly  or  indirectly,  is  absolutely  prohib- 
ited,” and  you  will  make  no  survey  in  which  you  are  interested. 

36.  Be  particular  to  write  all  proper  names  plainly. 

37.  Deputies  changing  their  residence  should  notify  this  office  of  such 
change. 

38.  I shall  expect  you  to  make  yourself  thoroughly  familiar  with  all  the 
mining  laws,  National  and  State,  as  well  as  with  these  instructions,  and  I am 
sure  that  errors  will  be  less  frequent,  and  this  office  as  well  as  yourself  will 
be  relieved  of  much  annoyance. 

39.  All  official  communications  must  be  addressed  to  the  Surveyor  Gen- 
eral, and  not  to  clerks  in  his  office. 

Very  respectfully, 

Albert  Johnson, 

Surveyor  General  of  Colorado. 


Sample  Field  Notes  furnished  Deputies. 

Survey  No.  500,  District  No.  3. 

FIELD  NOTES 

Of  the  survey  of  the  claim  of 

John  H.  Marshall, 

upon  the Excelsior Lode, 

in California. Mining  District, 

Lake  County,  Colorado. 


Surveyed  by George  D.  Williams, 

U.  S.  Deputy  Mineral  Surveyor. 

Survey  begun  October  23d 

and  completed “ 28th, 


.1881. 

1881. 


26 


ELEMENTS  OF  SURVEYING. 


[APP.  C. 


Feet. 


300. 


92. 

224.27 

952.72 

1256.36 

| 1500. 


Description  of  Survey. 

Beginning  at  Cor.  No.  1,  a spruce  post,  5 ft.  long,  4 ins. 
diam.,  set  2 ft.  in  the  ground,  with  mound  of  stone, 
marked  5J0  whence  the  W.  cor.  in  sec.  10,  T.  8 S., 
R.  80  W.  of  the  6tli  Prin.  Mer.  bears, 

S.  84°  26'  W.  1847.  4 ft. 

A spruce,  6 ins.  diam.,  marked  B.  T.  5J0,  bears  S.  78°  32' 
W.  32.9  ft. 

Thence  N.  11°  53'  E. 

Var.  14°  15'  E. 

To  Cor.  No.  2. 

A spruce  post  4 ft.  long,  4 ins.  dia.,  set  2 ft.  in  ground, 
with  mound  of  earth  marked  5g0,  whence, 

Holy  Cross  Mt.  bears  N.  41°  37'  W. ; 

Mt.  Leon  bears  N.  14°  29'  W. 

(No  other  bearings  available.) 

Thence  S.  78°  7'  E. 

Var.  14°  15'  E. 

To  road  running  S.  Easterly. 

Intersect  line  4-1  of  Sur.  No.  495,  Treasury  lode,  John 
P.  Jones  et  al.  claimants,  N.  15°  E.  79.91  'ft.  from  Cor. 
No.  4. 

Intersect  line  4-1  of  Sur.  No.  227,  Silver  Dollar  Lode, 
Henry  J.  Smith,  claimant,  at  S.  3°  W.  401.02  ft.  from  Cor. 
No.  1. 

Intersect  line  2-3  of  Sur.  No.  227,  S.  3°  W.  447  9 ft.  from 
Cor.  No.  2. 

To  Cor.  No.  3. 


APP.  C.] 


SAMPLE  FIELD  NOTES. 


27 


Feet. 


149.5 

198. 

300. 


112. 

196.75 

500.39 

911. 

1500. 


A granite  stone  25  x 11  x 7 ins.,  set  1 ft.  in  ground, 
with  mound  of  stones,  chiseled  6g0,  whence  a spruce 
6 ins.  dia.,  marked  B.  T.  5g0,  bears  S.  80°  23'  E.  20.4  ft.; 
B.  R.  5§0  chiseled  on  rock  in  place,  bears  N.  44°  E. 
30  ft. 

Thence  S.  11°  53'  W. 

Var.  14°  55'  E. 

Intersect  line  3-4  of  Sur.  No.  495,  at  N.  75°  W.,  218  ft. 
from  Cor.  No.  3. 

To  Gulch ; course  N.  E. 

To  Cor.  No.  4. 

A quartz  stone  24  x 13  x 8 ins.,  set  1 ft.  in  ground,  and 
mound  of  stone,  chiseled  5J0,  whence  B.  R.  5*0  chiseled 
on  large  boulder,  bears  N.  38°  22'  W.  25.4  ft. 

Mt.  Leon  bears  N.  19°  22'  W. 

Sherman  Peak  bears  S.  84°  45'  E. 

Cor.  No.  3,  Sur.  No.  495,  bears  N.  69°  23'  E.  258  ft. 

Cor.  No.  3,  Sur.  227,  bears  S.  18°  9'  W.  743.9  ft. 

Thence  N.  78°  7'  W. 

Var.  14°  55'  E. 

To  Gulch. 

Intersect  line  2-3  of  Sur.  No.  227,  N.  3°  E.,  748.45  ft.  from 
Cor.  No.  3. 

Intersect  line  4-1  of  Sur.  No.  227,  N.  3°  E.,  795.34  ft.  from 
Cor.  No.  4. 

To  road. 

To  Cor.  No.  1,  place  of  beginning. 


28 


ELEMENTS  OF  SURVEY  I NO. 


[app.  C. 


Area. 


Containing 10.32  acres. 

Less  conflicts  with..  

Sur.  No.  227=2.12  acres 

“ “ 495=2.4(5  “ 4.58  “ 

Leaving  net  area 5.74  acres. 


Location  . 

This  survey  is  located  in  the  N.  % and  S.  E.  U of  Sec.  10,  in  T 8 S R 
80  W. 

Improvements 

Upon  this  claim  consist  of : 

A discovery  shaft,  5'  x 3',  275  ft.  deep,  timbered,  which  bears  from  Cor 
No.  1,  N.  64  59'  E.  250.6  ft. 

A log  shaft-house,  56'  x 40',  the  S.  W.  corner  of  which  bears  from  Cor 
No.  1,  N.  65°  26'  E.  207  ft. 

A shaft  4'  x 4',  25  ft.  deep,  which  bears  from  Cor.  No.  1,  N.  88°  39  E. 
765  ft. 

An  open  cut  3'  x 10',  which  bears  from  Cor.  No.  3,  S.  58°  45'  W.  264  ft. 

1 hereby  certify  that  the  value  of  said  improvements,  together  with  the 
labor  expended  on  the  said  claim  by  the  claimant  and  his  grantors,  is  not 
less  than  Five  Hundred  Dollars  ($500). 

Improvements  made  by  other  parties  are  as  follows  : 

A shaft  4'  x 6',  78  feet  deep,  which  bears  from  Cor.  No.  3,  N.  83°  27'  W. 
389  ft. 


Adjoining  Claims. 

Surveys  Nos.  495  and  227.  No  others  known. 

Instruments  Used. 

A Gurley  Mountain  Transit,  with  solar  attachment  (or  the  instrument 
used),  and  steel  tape  (or  the  measure  used). 

All  courses  determined  bv  the  use  of  the  solar  apparatus. 

Address  op  Applicant, 

JOHN  H.  MARSHALL, 

P.  O.  Box  743,  Leadville,  Colo. 


[Then  follow  a list  of  the  assistants  employed  in  making  the  survey,  with 
their  affidavit,  as  to  its  conformity  to  instructions,  the  affidavit  of  the  Deputy 
Mineral  Surveyor,  and  the  certificate  of  the  Surveyor  General.] 


APP.  C.]  SAMPLE  PLAT.  29 


Survey  No.  500. 
Mineral  District  No  3. 

PLAT  of  the  claim  of  John  H.  Marshall  upon  the  Excelsior  Lode, 

California  Mining  District, Lake County,  State 

of  Colorado  Containing  an  area  of. . . .5.74. . . .acres.  Scale  of. . . . 
200  . feet  to  the  inch.  Variation.  ..  .East.  Surveyed  by  Geo.  D. 

Williams,  U.  S.  Deputy  Mineral  Surveyor. 

October  28,  1881. 


The  Original  Field  Notes  of  the  Survey  of  the  Claim  of 


upon  the from  which  this  plat  has  been  made,  have  been  examined 

and  approved,  and  are  on  file  in  this  office  ; and  I hereby  certify  that  they  furnish  such  an 
accurate  description  of  said  Mining  Claim  as  will,  if  incorporated  into  a patent,  serve  fully 
to  identify  the  premises,  and  that  such  reference  is  made  therein  to  natural  objects  and  per- 
manent monuments  as  will  perpetuate  and  fix  the  locus  thereof.  I further  certify  that  the 
value  of  the  labor  and  improvements  placed  thereon  by  the  applicant,  or  his  grantors,  is  not 
less  than  Five  Hundred  Dollars,  and  that  said  improvements  consist  of : 


as  appears  by  the  report  of  the  Deputy  Surveyor.  And  I further  certify  that  this  is  a correct 
Plat  of  said  Mining  Claim,  made  in  conformity  with  said  field  notes  of  the  survey  thereof. 

U S.  Surveyor  General's  Office, 

Denver,  Colorado. 

18... 


U.  S.  Surveyor  General  for  Colorado. 


OM!Vasmr  &i  UilBSU 


A TABLE 

OP 

LOGARITHMS  OF  NUMBERS. 


Remark. — In  the  following  table,  in  the  nine  right  hand 
columns  of  each  page,  where  the  first  or  leading  figures  change 
from  9’s  to  0’s,  points  or  dots  are  introduced  instead  of  the 
0’s,  to  catch  the  eye,  and  to  indicate  that  from  thence  the 
two  figures  of  the  Logarithm  to  be  taken  from  the  second 
column,  stand  in  the  next  line  below. 


2 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 D*  ' 

100 

101 

102 

103 

104 

105 

1 06 

\ol 

109 

I 10 

MI 

I I a 

1 1 3 

114 

1 1 5 

116 

000000 
4321 
8600 
012837 
7033 
021189 
53  06 
o384 
o33434 
7436 

o4i393 
5323 
02 1 8 

053078 

69o5 

060698 

4458 

0434 

4751 

0026 
3259 
745 1 
i6o3 
5715 

& 

0868 

5i8i 

945i 

3680 

7868 

2016 

6i25 

•195 

4227 

8223 

2182 

6io5 

$?6 

7666 

1452 

5206 

i3oi 

5609 

9876 

1734 

6o38 

•3oo 

4521 

8700 

2841 

6942 

1004 

5o29 

9017 

6?8? 

2166 

6466 

•724 

494o 

liSt 

735o 

1408 

543o 

9414 

3362 

7$ 

4996 

88o5 

2598 

6894 

1147 

536o 
953 1 
3664 
7757 
1812 

3029 

7321 

1570 

5779 

994] 

4075 

1 3461 

1 7748 
>993 
6197 
•36 1 
4486 
8571 
2619 
6629 
•602 

4540 

8442 

2309 

6142 

0942 

37°o 

7443 

3891 

8174 

24i5 

6616 

•775 

4896 

43* 
428> 
424 
419 
416 
412 
408 
] 404 

4oo 

396 

393 

389 

386 

382 

379 

376 

372 

4100 

8284 

2428 

6533 

•600 

4628 

8620 

2576 

6495 

•38o 

423o 

8164 

2216 

6230 

•207 

4148 

8o53 

i9a4 

5760 

9563 

3333 

7071 

8978 

3021 

7028 

•998 

4932 

883o 

583o 

9811 

3755 

7664 

1 538 
5378 
9185 
2958 
6699 

7825 

1787 

5]  14 

9606 

3463 

7286 

io]5 

4832 

•766 
401 3 
8426 
2206 

5953 

2694 

6524 

•320 

4o83 

78i5 

8046 

1829 

558o 

2582 

6326 

lll 

8186 

8557 

8928 

9208 

9668 

3352 

••38 

•40] 

•776 

1 U5 

1 5 1 4 

369 

118 

071882 

2250 

26l] 

2985 

3718 

4o85 

445i 

4816 

5i  82 

366 

M9 

5547 

59i  2 

6276 

6640 

7004 

7368 

773i 

8094 

8457 

8819 

363 

120 

079181 

082785 

0543 

gg 

•266 

•626 

•087 

45]6 

1 347 

1707 

2067 

2426 

36o 

I 2 1 

3144 

386 1 

4219 

4934 

5a9It 

5647 

6004 

& 

122 

63  60 

6716 

7071 

7426 

7]8i 

8i36 

8490 

8845 

9198 

o552 

123 

090  5 
093432 

•258 

*6l  I 

•963 

i3  1 5 

1667 

2018 

2370 

2721 

3o7i 

35i 

134 

3772 

4122 

44]i 

4820 

6169 

55i8 

5866 

62i5 

6562 

349 

125 

6010 

725] 

7604 

795i 

8298 

8644 

8900 

2434 

9335 

968 1 

••26 

346 

126 

100371 

0715 

io59 

i4o3 

1747 

2091 

2777 

3 1 19 

3462 

343 

,2I 

38o4 

4146 

4487 

4828 

5i69 

55io 

585 1 

6191 

653 1 

68ji 

340 

128 

7210 

7549 

7888 

8227 

8565 

89o3 

9241 

9579 

3$ 

•233 

338 

129 

1 10590 

0926 

1263 

1 599 

1934 

2270 

26o5 

2940 

3609 

335 

i3o 

11 3943 

$3 

4611 

4944 

5278 

56ii 

5943 

6276 

9586 

6608 

6940 

333 

1 3 1 

7271 

7934 

8265 

8595 

1888 

8926 

9266 

3$ 

6456 

•245 

33o 

1 3 a 

1205^ 

0903 

I 23  I 

i56o 

2216 

2544 

2871 

3525 

328 

1 33 

38$2 

4178 

45o4 

483o 

5i  56 

5481 

58o6 

6i3i 

6781 

325 

1 34 

7io5 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

••12 

323 

i35 

i3o334 

o655 

0977 

1298 

1619 

2260 

258o 

2900 

3219 

640J 

321 

1 36 

3539 

3858 

4496 

4814 

545 1 

5769 

6086 

3i8 

,37 

6]2I 

7o37 

767 1 

7987 

83o3 

8618 

8934 

9249 

9564 

3 1 5 

i3o 

0879 

•194 

•5o8 

•822 

1 136 

i45o 

1763 

4885 

2076 

2389 

2702 

3 14 

1 39 

i43oi5 

3327 

3639 

3961 

4263 

4574 

5i96 

5507 

58i8 

3 1 1 

140 

146128 

6438 

6748 

9835 

7o58 

7367 

7676 

7985 

8294 

86o3 

8911 

309 

141 

9527 

•142 

•449 

•756 

io63 

1370 

1676 

1982 

3oi 

3o5 

143 

2^94 

2900 

32o5 

35io 

38i5 

4120 

4424 

4728 

5o32 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7i54 

7457 

7759 

8061 

3o3 

144 

8362 

8664 

8965 

9266 

9567 

9868 

•168 

•469 

•760 

1068 

3oi 

145 

i6i368 

1667 

1967 

2266 

2564 

2863 

3i6i 

3460 

3758 

6726 

4o55 

299 

146 

4353 

465o 

4947 

7008 

5244 

5541 

5838 

6i34 

6430 

7022 

297 

295 

'47 

7317 

7613 

82o3 

8497 

8792 

9086 

9380 

9674 

9968 

14- 

170262 

o555 

0848 

1141 

1434 

1726 

2019 

a3i  1 

26o3 

2895 

293 

149 

3 1 86 

3478 

3769 

4060 

435 1 

4641 

4932 

5222 

55i2 

58o2 

291 

i5o 

176091 

638i 

6670 

& 

7248 

7536 

7825 

81 13 

8401 

8689 

1 558 

289 

1 5i 

8077 

9264 

9552 

•126 

•4i3 

•985 

1272 

28] 

1 5a 

181844 

2129 

241,5 

2700 

2985 

3270 

3839 

4123 

4407 

285 

1 53 

4691 
752 1 ' 

4075 

5259 

5542 

5§25 

6108 

639i 

6674 

6956 

7a39 

283 

.54 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

2289 

s; 

••5 1 

281 

1 55 

190332, 

0612 

0892 

H7i 

U5i 

i]3o 

2010 

2846 

279 

i56 

3 1 25! 

34o3 

368 1 

3909 

423] 

4514 

5069 

5346 

5623 

278 

*57 

a? 

6176 

6453 

6729 

7006 

7281 

7832 

8107 

8382 

276 

i58 

8082 

9206 

948i 

9755 

••29 

•3o3 

•577 

•85o 

1124 

274 

159 

201 397 

1070 

1943 

2216 

2488 

2761 

3o33 

33o5 

3577 

3848 

272 

N. 

0 1 

1 

3 

4 

5 

6 

7 

8 

9 

D. 

A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


3 


r 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

IX 

160 

204120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

1 6 1 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

^79 

9247 

269 

162 

q5i5 

0783 

••5 1 

•3  IQ 

•586 

•853 

1121 

1 388 

1654 

1921 

267 

1 63 

212188  2454 

2720 

2986 

3252 

35i8 

3783 

4049 

43i4 

4579 

266 

164 

4844 

5ioq 

5373 

5638 

5902 

6166 

643o 

6694 

6957 

j 7221 

264 

i65 

7484  7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

, 9846 

262  1 

166 

2 10108 

0370 

o63i 

0892 

1 1 53 

1414 

1675 

i936 

2196 

2456 

261  I 

,61 

2716 

2976 

3236 

3496 

3755 

401 5 

4274 

4533 

4792 

5o5i 

25a  j 

166 

53o9 

5568 

5826 

6084 

6342 

6600 

6858 

7 1 1 5 

7372 

763o 

258 1 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

•193 

256] 

*7° 

230449 

0704 

0960 

1 2 1 5 

1470 

1724 

1979 

2234 

2488 

2742 

254 

171 

2996 

325o 

35o4 

3757 

401 1 

4264 

4517 

4770 

5o23 

5276 

253 

172 

5528 

578i 

6o33 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

i73 

8046 

8297 

8548 

8799 

9049 

9299 

955o 

9800 

••5o 

•3oo 

1 250 

174 

240649 

0799 

1048 

1 297 

1 546 

J795 

2044 

2293 

2641 

2790 

249 

175 

3o38 

3286 

3534 

3782 

4o3o 

4277 

4525 

4772 

5oi9 

5266 

248! 

176 

55i3 

5759 

6006 

6252 

6499 

6745 

699 1 

7237 

7482 

7728 

246 

«77 

7973 

8219 

8464 

8709 

8954 

9J98 

9443 

9687 

9o32 

•176 

245 

178 

250420 

0664 

0008 

1 1 5 1 

1396 

1 638 

1881 

2125 

2368 

2610 

243 

179 

2853 

3096 

3338 

358o 

3822 

4064 

43o6 

4548 

4790 

5o3i 

242 

180 

255273 

55i4 

5755 

5996 

6237 

6477 

6718 

6g58 

7198 

7439 

241 

181 

7679 

79l8 

8i58 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

26007 1 

o3io 

o548 

0787 

1025 

1263 

i5oi 

1739 

1 976 

2214 

238 

1 83 

245 1 

2688 

2925 

3x62 

3399 

3636 

3873 

4109 

4846 

4582 

237 

184 

4818 

5o54 

5290 

5525 

5761 

6996 

6232 

6467 

6702 

6937 

235 

1 85 

7172 

74o6 

7641 

7875 

8lIO 

8344 

8578 

8812 

9046 

9279 

234 

186 

95i3 

9746 

9980 

•2 1 3 

•446 

•679 

•912 

1144 

1377 

1609 

233 

i82 

271842 

2074 

23o6 

2538 

2770 

3ooi 

3233 

3464 

3696 

3927 

232 

188 

41 58 

4389 

4620 

485o 

5o8i 

53 11 

5542 

5772 

6002 

6232 

23o 

189 

6462 

6692 

6921 

7i5i 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278764 

8982 

9211 

943q 

9667 

9895 

•123 

•35 1 

•578 

•806 

228 

191 

28io33 

1261 

1488 

1716 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

33oi 

3527 

3753 

3979 

42o5 

443 1 

4656 

4882 

6107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6906 

7i3o 

7354 

7578 

225 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

958o 

9812 

223 

195 

290035 

0267 

0480 

0702 

0925 

1147 

1369 

1 59i 

1813 

2o34 

222 

196 

2256 

2478 

2699 

2920 

3i4i 

3363 

3584 

38o4 

4025 

4246 

221 

■97 

4466 

4687 

4907 

5i  27 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

*99 

8853 

9071 

9289 

9507 

9725 

9943 

•161 

•378 

•595 

•8i3 

2l8 

200 

3oioJo 

1247 

1464 

1681 

1898 

2114 

233 1 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4009 

4275 

4491 

4706 

4921 

5i36 

2l6 

202 

535 1 

5566 

578i 

6996 

6211 

6425 

6609 

6854 

7068 

7282 

2 1 5 

203 

7496 

7710 

7924 

8137 

835 1 

8564 

8778 

8991 

9204 

9417 

213 

204 

963o 

9843 

••56 

•268 

•481 

•693 

•906 

1118 

i33o 

1 542 

212  1 

2o5 

311754 

1966 

2177 

2389 

2600 

2812 

3o23 

3234 

3445 

3656 

211 ; 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5i3o 

5340 

555i 

5760 

210; 

2°T  | 

5970 

6180 

6390 

6699 

6809 

7018 

7227 

7436 

7646 

7854 

209! 

108  ! 

8o63 

8272 

8481 

8689 

8898 

9106 

93i4 

9622 

973° 

9938 

208 

209  i 

320146 

o354 

0662  1 

0769 

0977 

1 184 

1391 

1598 

i8o5 

2012 

207 

210  | 

322219 

2426 

2633 

2839 

3o46 

3252 

3458 

3665 

387i 

4077 

206 

21 1 

4282 

4488 

4694 

4899 

5io5 

53io 

55i6 

5721 

5926 

6i3i 

205 

212 

6336 

654i 

6745 

69D0 

7i55 

735o 

7563 

7167 

7972 

8176 

204 

3 1 3 

838o' 

8583 

8787 

899 1 

9>94 

9398 

9601 

9805 

•••8 

•211 

203 

214 

33o4i4  0617 

0819 

1022 

1225 

1427 

i63o 

i832 

2o34 

2236 

202 

2 1 5 

2438 

2640 

2842 

3 044 

3246 

3447 

3649 

385o 

4o5i 

4253 

202 

216 

4454 

4655 

4856 

5o57 

5257 

5458 

5658 

5859 

6069 

6260  1 

201 

7ll 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858  | 

8o58 

8257  1 

200 

218 

8456 

8656 

8855 

9054 

9253 

945 1 

9650 

9849  | 

••47 

•246 

I99 

219 

340444, 

0642 

0841  1 

1039 

1237 

1435 

i632 

i83o  1 

2028  ; 

2225  ' 

198 

N. 

0 1 

• 

1 

3 

4 

5 

6 

7 j 

8 

]>£ 

15 


1 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


N. 

0 

■ 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

210 

342423 

2620 

iSii 

3oi4 

3212 

3409 

3 606 

3 802 

3999 

4196 

197 

221 

4392 

4689 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6167 

196 

222 

223 

6353 

83o5 

6549 

85oo 

6744 

8694 

6939 

8889 

7135 

9083 

733o 

9278 

7525 

9473 

7720 

9666 

79*5 

9860 

81 10 

••54 

i95 

194 

224 

350248 

0442 

o636 

0829 

1023 

1216 

1410 

i6o3 

1796 

1989 

iq3 

225 

2i83 

2375 

2568 

2761 

2954 

3i47 

3339 

3532 

3724 

39i6 

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226 

4108 

43oi 

4493 

4685 

4876 

5o68 

5260 

5452 

5643 

5834 

192 

227 

22& 

6026 

7935 

6217 

8125 

6408 

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6599 

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6981 

8886 

I 7*72 

1 9076 

7363 

9266 

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9456 

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191 

190 

229 

9835 

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j *972 

1161 

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1 539 

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361728 

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2105 

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3o48 

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232 

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187 

233 

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7542 

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8101 

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186 

234 

9216 

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9587 

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1 85 

235 

371068 

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184 

236 

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3096 

3280 

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383 1 

401 5 

4198 

4382 

4565 

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257 

4748 

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5298 

5481 

5664 

5846 

6029 

6212 

6394 

1 83 

238 

239 

S& 

6942 

8761 

7124 

8943 

7306 

9124 

7488 

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767° 

9487 

7852 

9668 

8o34 

9849 

8216 

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182 

181 

240 

380211 

0392 

o573 

0754 

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1 1 1 5 

1296 

1476 

1656 

1837 

181 

241 

2017 

2197 

2377 

2557 

2737 

29*7 

3097 

3277 

3456 

3636 

180 

242 

38 1 5 

3996 

4174 

4353 

4533 

4712 

4891 

5070 

6249 

5428 

179 

243 

56o6 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

244 

7890 

7068 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

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245 

9166 

9343 

9520 

9698 

9875 

••5 1 

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•406 

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246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

2697 

2873 

3o48 

3224 

3400 

3575 

3926 

4101 

4277 

176 

248 

4452 

4627 

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4977 

5i  52 

5326 

55oi 

5676 

585o 

6025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

25o 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

25i 

9674 

9847 

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1228 

173 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

26o5 

2777 

2949 

172 

253 

3 1 2 1 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

Hi 

254 

4834 

5oo5 

5»76 

5346 

55i7 

5688 

5858 

6029 

6199 

6370 

Hi 

255 

654o 

6710 

6881 

jo5i 

7221 

73qi 

756i 

773i 

79° 1 

8070 

110 

256 

8240 

8410 

8579 

8749 

8918 

9087 

2**7 

9426 

95q5 

9764 

169 

257 

9933 

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•271 

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•600 

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1114 

1283 

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160 

258 

41 1620 

1788 

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2124 

2298 

2461 

2629 

2796 

2964 

3 1 32 

168 

259 

33oo 

3467 

3635 

38o3 

3970 

4i37 

43o5 

4472 

4639 

4806 

167 

260 

414973 

5i4o 

5307 

5474 

564i 

58o8 

5974 

6141 

63o8 

6474 

l67 

261 

6641 

6807 

6974 

7149 

7306 

7472 

7638 

7804 

7970 

8i35 

166 

262 

83oi 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

962s 

979 1 

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263 

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1 1 10 

1275 

1449 

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264 

421604 

1788 

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2097 

2261 

2426 

2690 

2754 

2918 

3082 

164 

265 

3246 

3410 

3574 

3787 

39oi 

4o65 

4228 

4392 

4555 

47 18 

164 

266 

4882 

5o45 

5208 

5871 

5534 

6697 

586o 

6023 

6186 

6849 

1 63 

267 

65 1 1 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7974 

162 

26b 

81 35 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

959 1 

162 

269 

9752 

99i4 

••76 

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1042 

1203 

161 

270 

43 1 364 

1 525 

1 685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3i3o 

3290 

345o 

36io 

377° 

393o 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159 

273 

274 

6i63 

7751 

6322 

7909 

6481 

8067 

6640 

8226 

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6957 

8542 

7116 

8701 

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7433 

9017 

7592 

9175 

3 

275 

276 

9^33 

440909 

9401 

1066 

9648 

1224 

9806 

1 38i 

•122 

1695 

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•437 

2°°9 

•5o4 

2166 

•i 152 

2823 

1 58 

1 57 

277 

2480 

2637 

2793 

2950 

3 106 

3263 

3419 

3576 

3732 

3889 

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278 

4045 

4201 

4357 

45 1 3 

4669 

4825 

4981 

5i37 

5293 

5449 

1 56 

1 55 

279 

56o4 

6760 

5916 

6071 

6226 

6382 

6537 

6692 

6848 

7008 

N. 

0 

« 

2 

3 

4 

5 

6 

7 

« i 

9 

D. 

A TABLE  OK  LOGARITHMS  FROM  1 TO  1C, 000. 


6 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

1 9 

D. 

280 

447 1 58 

73 1 3 

7468 

7623 

7778 

7933 

8088 

8242 

83v7 

8552 

1 55 

281 

8706 

8861 

901 5 

9170 

9324 

9478 

9633 

9787 

994* 

1 54 

282 

450249 

1 o4o3 

o55j 

0711 

o865 

1018 

1172 

i326 

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i633 

1 54 

283 

1786 

1940 

2090 

12247 

2400 

2553 

2706 

2859 

3012 

3 1 65 

i5J 

284 

33i8 

3471 

3624 

3777 

3g3o 

4082 

4235 

4387 

4540 

4692 

1 53 

285 

4845 

4997 

5i5o 

53o2 

5454 

56o6 

5758 

5910 

6062 

, 6214 

*5i 

286 

6366 

65i8 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

773* 

n5> 

287 

7882 

8o33 

8184 

8336 

8487 

8638 

8789 

8940 

9°9* 

9242 

i5i 

280 

9392 

9543 

9694 

9845 

9995 

•146 

•296 

•447 

•597 

•748 

i5i 

289 

460898 

1048 

1198 

i348 

M99 

1649 

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1948 

2098 

2248 

1 i5o 

290 

462398 

2548 

2697 

2847 

2997 

3i46 

3296 

3445 

3594 

3744 

i5o 

291 

3893 

4042 

4191 

434o 

4490 

4639 

4788 

4936 

5o85 

5234 

149 

292 

5383 

£532 

568o 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

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293 

6868 

7016 

7*54 

7312 

746o 

7608 

7756 

7904 

8o52 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9627 

9675 

148 

295 

9822 

9969 

•116 

•263 

•410 

•55t 

•704 

•85 1 

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ii45 

l47 

296 

471292 

1438 

1 585 

1732 

1878 

2020 

2171 

23 1 8 

2464 

2610 

146 

$ 

2706 

4216 

2903 

4362 

3049 

45o8 

3195 

4653 

3341 

4799 

3487 

4944 

3633 

5090 

£$ 

3925 
538 1 

4071 

5526 

146 

146 

299 

5671 

58i6 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

3oo 

477121 

7266 

74ii 

7555 

7700 

7844 

7989 

81 33 

8278 

8422 

145 

3oi 

8566 

8711 

8855 

°999 

9143 

9287 

943* 

9575 

97*9 

9863 

144 

302 

480007 

oi5i 

0294 

0448 

0682 

0725 

0869 

1012 

1 1 56 

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*44 

(_i°3 

1443 

1 586 

1729 

1872 

2016 

2169 

23o2 

2445 

2588 

2731 

143 

3o4 

2874 

3oi6 

3 1 59 

33o2 

3445 

3587 

3730 

3872 

401 5 

4*57 

143 

3o5 

43  00 

4442 

4585 

4727 

4869 

5oi  1 

5i  53 

5295 

5437 

5579 

142 

3o6 

5721 

5863 

6oo5 

6147 

6289 

643o 

65t2* 

6714 

6855 

6997 

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3°7 

7 1 38 

7280 

7421 

7563 

?7°4 

7845 

7086 

8127 

8269 

8410 

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3o8 

855 1 

8692 

8833 

8974 

9**4 

9255 

9396 

9537 

9677 

9818 

14* 

309 

9958 

•*99 

•239 

•38o 

•520 

•661 

•801 

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1081 

1222 

140 

3io 

49i362 

l5o2 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

3 1 1 

2760 

2900 

3o4o 

3179 

3319 

3458 

3 597 

3737 

3876 

401 5 

139 

3l2 

4i55 

4294 

4433 

4572 

47*1 

485o 

4989 

5128 

5267 

54o6 

139 

3i3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

65i5 

6653 

6791 

1 39 

3i4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8o35 

8173 

i38 

3 1 5 

83 11 

8448 

8586 

8724 

8862 

8999 

9i37 

9275 

9412 

955o 

1 38 

3i6 

9687 

9824 

9962 

•#99 

•236 

•374 

•5i  1 

•648 

•785 

•922 

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3*7 

5oio59 

1196 

1 333 

1470 

1607 

1744 

1880 

2017 

2i54 

2201 

i37 

3i8 

2427 

2564 

2700 

2837 

2073 

3109 

3246 

3382 

35i8 

3655 

1 36 

319 

3791 

3927 

4o63 

4199 

4335 

447* 

4607 

4743 

4878 

5oi4 

1 36 

320 

5o5i5o 

5286 

5421 

5557 

5693 

5828 

5964 

6099 

6234 

6370 

1 36 

321 

65o5 

6640 

6776 

6911 

7046 

7181 

7J16 

745i 

7586 

7721 

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322 

7856 

799 1 

8126 

8260 

8395 

853o 

8664 

8799 

8934 

9068 

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323 

9203 

9337 

9471 

9606 

9740 

9874 

•••9 

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1 34 

324 

5io545 

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0947 

1081  ' 

x 2 1 5 

1 349 

1482 

1616 

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1 34 

325 

1 883 

2017 

2l5l 

2284 

2418  ; 

255i 

2684 

2818 

2951 

3o84 

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326 

3218 

335i 

3484 

3617 

375o 

3883 

4016 

4*49 

4282 

44i4 

1 33 

32*7 

4548 

4681 

4813 

4946 

5°79  ! 

5211 

5344 

5476 

5609 

5741 

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328 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

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329 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

825i 

8382 

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33o 

5i85i4 

8646 

8777 

8909 

9040 

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93o3 

9434 

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9697 

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33 1 

9828 

9959 

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•61 5 

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1007 

1 3 1 

332 

5a i i38 

1269 

1400  | 

i53o 

1661 

1792 

1922 

2o53 

2 1 83 

a3 1 4 

1 3 1 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

36i6 

i3o 

334 

3746 

3876 

4006  | 

4i36 

4266 

4396 

4526 

4656 

4785 

49*5 

i3o 

335 

5o45 

5i74 

53o4  ! 

5434 

5563 

5693 

5822 

5951 

6081  ! 

6210 

129 

336 

337 

6339 

7630 

6469 

775q 

6598  | 
7888  ! 

6727 

8010 

6856 

8i45 

6985 

8274 

7114 

8402 

7243 
853 1 

7372 

8660 

75oi 

8788 

129 

129 

338 

339 

.fi'l 

53o2oo 

9045 

0328 

9*74  , 
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9302 

o584 

943o 

0712 

9559 

0840 

$2 

98 1 5 

109^*  | 

9943 

1223 

••72 

i35i 

126 

120 

N. 

0 

1 

2 

3 

4 

5 

6 

HZi 

8 

— L 

9 

D. 

6 


a TABLE  OF  LOGARITHMS  FROM  1 TO  10,000 


N. 

0 

■ 

2 

3 

4 

5 

6 

7 

1 » 

v 

34o 

531479 

1607 

1734 

1862 

•‘99° 

2117 

2245 

2372 

25oo 

2627 

3899 



128 

341 

2754 

2882 

3009 

3 1 36 

3264 

3391 

35i8 

3645 

3772 

127 

343 

4026 

41 53 

428o 

4407 

4534 

4661 

ss 

! 4914 

5o4i 

5167 

X2? 

343 

5294 

5421 

5547 

5674 

6987 

58oo 

5927 

6180 

63o6 

6432 

126 

344 

6558 

6685 

68l  I 

7063 

7i89 

73 1 5 

744i 

126 

345 

7819  7945 

8071 

8197 

8322 

8448 

8574 

8899 

99^4 

126 

346 

9076 

9202 

9327 

9452 

JS£ 

9703 

9829 

3s 

•204 

125 

347 

5 40329 

0455 

o58o 

0705 

0955 

1080 

1205 

1454 

125 

34» 

1579 

1704 

1829 

jm53 

2078 

22o3 

23a7 

2452 

2576 

2701 

125 

349 

?825 

2950 

3074 

3199 

3323 

3447 

357i 

3696 

3820 

t944 

124 

35o 

d 44068 

4192 

43 1 6 

444o 

4564 

4688 

4812 

4936 

1 5o6o 

5x83 

124 

35i 

5307 

543i 

5555 

6678 

58o2 

5925 

6049 

6172 

6296 

7529 

6419 

124 

35a 

6543 

6666 

6789 

6913 

7036 

7 1 59 

7282 

7405 

7662 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

85i2 

8635 

8758 

8881 

123 

354 

9oo3 

9126 

9249 

9871 

9494 

9616 

9739 

9861 

9984 

•106 

123 

355 

550228 

o35i 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

i328 

122 

356 

1460 

1572 

1694 

1816 

1938 

2060 

2181 

23o3 

2425 

2547 

122 

35  7 

2668 

2790 

2911 

3o33 

3 1 55 

3276 

3398 

35i9 

364o 

3762 

1 2 1 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

473i 

4852 

££ 

I 2 1 

359 

5094 

52 1 5 

5336 

5457 

5578 

5699 

5820 

6940 

6061 

I 21 

36o 

5563g3 

6423 

6544 

6664 

6786 

6905 

7026 

7146 

8340 

7267 

l ?387 

120 

36i 

J5o1 

8709 

7627 

7748 

7868 

7988 

8108 

8228 

8469 

1 8589 

120 

362 

8829 

8948 

9068 

9188 

93o8 

9428 

9548 

9787 

120 

363 

99°7 

••26 

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364 

56iioi 

1221 

1 34o 

1459 

1578 

1698 

1817 

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2o55 

2174 

1 19 

365 

2293 

2412 

253i 

265o 

2769 

2887 

3oo6 

3i  25 

3244 

3362 

1 19 

II9 

366 

3481 

36oo 

3718 

3837 

3955 

4074 

4192 

43xi 

4429 

4548 

367 

4666 

4784 

4903 

5021 

5x39 

5267 

5376 

5494 

5612 

5730 

1x8 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

1 18 

369 

7026 

7*44 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

37o 

568202 

83i9 

8436 

8554 

8671 

8788 

89o5 

9023 

9140 

9257 

l!7 

37i 

9374 

9491 

9608 

9725 

9842 

9959 

•.76 

•309 

•426 

117 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

2523 

1476 

2639 

x5o2 

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373 

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2872 

1826 

1942 

2058 

2174 

3336 

2291 

2407 

2755 

110 

$ 

2988 

3io4 

3220 

3452 

3568 

3684 

3800 

39x5 

116 

4o3l 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5i88 

53o3 

5419 

5534 

565o 

5765 

588o 

5996 

61 1 1 

6226 

1 1 5 

fa 

634i 

mi 

6457 

7607 

6572 

7722 

6687 

7836 

6802 

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1 1 5 
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379 

8754 

8868 

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9212 

9326 

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9555 

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114 

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1089 

1 1 53 

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1495 

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114 

382 

2o63 

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2291 

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2631 

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2 97  2 

3o85 

1 14 

383 

3312 

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3652 

3765 

4896 

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4218 

1 1 3 

384 

4444 

4557 

4670 

4783 

6009 

5x22 

5235 

5348 

ix3 

385 

5461 

5574 

5686 

§8 

5912 

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6137 

6250 

6362 

6476 

1 13 

386 

6587 

6700 

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7037 

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7262 

7374 

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112 

387 

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8047 

8160 

8272 

8384 

8608 

1 12 

388 

8944 

9o56 

9167 

9279 

939 1 

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9726 

9838 

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389 

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i5io 

1621 

1732 

1843 

x955 

2066 

111 

391 

££ 

2288 

25io 

2621 

2732 

2843 

2954 

3064 

3175 

1 1 1 

392 

3397  j 
45o3 

36i8 

3840 

3950 

4061 

417* 

4282 

111 

393 

4393 

4614 

4724 

4945 

5o55 

5i65 

5276 

5386 

no 

394 

5496 

56o6 

57>7 

5827 

5937 

6047 

61  5t 

6a&7 

6377 

Ml 

IIO 

395 

6597 

7695 

87a1 

6707 

6817 

6927 

7o37 

7l4° 

7256 

736o  I 

7476 

7586 

1 10  I 

396 

7806 

7914 

8024 

Hi  34 

8243 

8353 

8462  | 

8672 

8681 

IIO 

397 

8900 

9009 

9ir9 

9228 

933I 

•428 

9446 

9556 

9665 

9774 

109 

396 

9883 

999 2 
1082 

•101 

•21 0 

•319 

•537 

•646 

•755 

•864 

I09 

399 

600973 

1 191 

| 1 299 

1408 

i5i7 

1626 

1734 

1843 

!95x 

109 

N. 

3 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

A TABLE  OP  LOGARITHMS  FROM  J TO  10,000. 


7 


N. 

0 

1 3 

3 

4 

5 

6 

7 

8 

1 

9 

1 D’ 

400 

602060 

2169 

2277 

2386 

1494 

a6o3 

2711 

2819 

2928 

3o36 

i .08 

401 

3144 

3253 

336i 

3469 

3686 

3794 

3902 

4010 

1 4118 

108 

40  a 

4226 

4334 

4442 

455o 

4658 

4766 

4874 

4982 

5089 

1 5i97 

1 108 

4o3 

53o5 

54i3 

55ai 

56a8 

5i3b 

5844 

5961 

6059 

6166 

6274 

1 108 

404 

S38i 

6489 

6596 

6704 

681 1 

6919 

7026 

ji33 

I 734i 

7348 

107 

4o5 

7455 

7562 

7669 

777  7 

7884 

7991 

1 8098 

8205 

83i  2 

8419 

107 

406 

8526 

8633 

8740 

8847 

8964 

9061 

9167 

9274 

938i 

9488 

1 i°7 

407 

9594 

9701 

9808 

ggU 

••21 

•128 

•234 

•34i 

•447 

•554 

1 107 

408 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

i4o5 

1 5 1 1 

1617 

I 106 

409 

1723 

1829 

1936 

2042 

2148 

2264 

a36o 

2466 

2572 

2678 

; 106 

410 

612784 

2890 

2996 

3l02 

3207 

! 33 13 

3419 

35a5 

363o 

3736 

106 

4i  1 
412 

3842 

4897 

4o53 

5io8 

4159 

5213 

4264 

53i9 

1 i3' 70 

1 5424 

4475 

5529 

458 1 
5634 

4686 

5740 

£8 

106 

io5 

4i3 

5g5o 

6o55 

6160 

6265 

6370 

1 6476 

658i 

6686 

67?° 

6895 

io5 

414 

415 

7000 

8048 

7io5 
01 53 

7210 

8a57 

i3i5 

8362 

7420 

8466 

! 7525 
8571 

2$ 

'2$ 

2$ 

io5 

io5 

416 

9093 

9198 

9302 

9406 

95ii 

9615 

9719 

9824 

9928 

••32 

104 

4l7 

620146 

0240 

o344 

0448 

o552 

o656 

0760 

0864 

0968 

1072 

104 

416 

1176 

1280 

1384 

1488 

1 592 

1695 

*799 

1903 

2007 

21 10 

104 

419 

2214 

a3i8 

2421 

a5a5 

2628 

2782 

2835 

2939 

3o42 

3146 

104 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

io3 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

io3 

42a 

53 1 2 

541 5 

55i  8 

5621 

5724 

5827 

6929 

6o3a 

61  3d 

6238 

io3 

423 

6340 

6443 

6546 

6648 

bj5i 

6853 

6g56 

7058 

7161 

7263 

io3 

424 

7366 

7468 

l5.11 

1115 

7878 

798o 

8082 

81 85 

°387 

102 

425 

838g 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

426 

94io 

g5i2 

9613 

97 1 5 

98i7 

9919 

••21 

•123 

•224 

•3a6 

102 

427 

630428 

o53o 

o63i 

0733 

o835 

o936 

io38 

1139 

1241 

i342 

102 

426 

1444 

i545 

1647 

1748 

1849 

1951 

2052 

2i53 

2255 

2356 

IO. 

429 

2467 

2559 

2660 

2761 

2862 

2963 

3064 

3i65 

3266 

3367 

IOl 

43o 

633468 

3569 

3670 

3771 

3872 

39I3 

4074 

4H5 

4216 

4376 

IOO 

43 1 

4477 

4578 

4679 

4779 

4880 

4981 

5o8i 

5182 

5283 

5383 

IOO 

432 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

IOO 

433 

434 

6488 

749° 

6588 

759° 

6688 

7600 

6789 

7700 

6889 

789? 

6989 

7900 

7089 

8090 

7i89 

8190 

7290 

8290 

2$ 

IOO 

99 

435 

8489 

8589 

mg 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

99 

436 

9486 

9586 

9686 

9785 

g885 

9984 

••84 

•i83 

•283 

•382 

99 

437 

64048 1 

o58i 

0680 

0779 

0879 

0978 

1077 

I!77 

1276 

i3i5 

99 

438 

439 

1474 

2465 

1573 

2563 

1672 

2662 

I?r 

2761 

1871 

2860 

1970 

2959 

2069 

3o58 

2168 

3 1 56 

2267 

3255 

2366 

3354 

99 

99 

440 

643453 

355i 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

98 

44i 

4439 

4537 

4636 

4734 

4832 

493 1 

5029 

5127 

5aa6 

5324 

98 

442 

5422 

55ai 

5619 

57H 

58i5 

5913 

6011 

6110 

6208 

63o6  1 

98 

443 

444 

445 

6404 

7383 

836o 

65oa 

7481 

8458 

6600 

llM 

6698 

6796 

2$ 

6894 

6992 

2$ 

d? 

9048 

9140 

7285 
8262  | 
9237  , 

98 

98 

97 

446 

9335 

9432 

953o 

9627 

9724 

9821 

99*9 

••16 

•1 1 3 

•210  1 

97 

447 

65o3o8| 

o4o5 

o5oa 

0599 

0696 

o793 

0890 

0987 

1084 

1181 

97 

448 

1278 

i375 

1472 

1569 

1666 

1762 

1809 

1956 

ao53 

2l5o  - 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3x  16  1 

97 

45o 

6532i3 

3309 

34o5 

35oa 

3598 

3695 

3791 

3888 

3984 

408c 

96 

45 1 

41771 

4273 

4369 

4465 

4562 

4658 

47?4 

485o 

4946 

5o42 

96 

452 

5i38 

5a35 

533 1 

5427 

5523 

5619 

57 1 5 

58io 

5906 

6002 

96 

453 

6098 

6194 

6290 

6386 

6482 

6677 

6673 

6769 

6864 

6960 

96 

454 

7066 

71 32 

7247 

7343 

7438 

7584 

7629 

7726 

7820 

79l6 

96 

455 

8011 

8107 

8202 

8ag8 

8393 

8488 

8584 

8679 

6774 

8870 

95 

456 

8965 

9060 

9i55 

9260 

9346 

9441 

9536 

9631 

9726 

9821 

95 

457 

9916 

**I  I 

•106 

•201 

•296 

•3a1 

•486 

•58 1 

•676 

•771 

95 

458 

66o865 

0960 

io55 

1 i5o 

1245 

1389 

1434 

1529 

i6a3 

1718 

95 

459 

i8i3 

1907 

2002 

2096 

2191 

2286 

j38o 

2473 

2569 

2663 

95 

N. 

4 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

8 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

460 

662758 

2852 

2947 

3 04 1 

3i35 

3a3o 

3324 

3418 

35 1 2 

3607 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

436o 

4454 

4548 

46a 

4642 

4736 

483o 

4924 

5oi8 

5 1 1 2 

5ao6 

5299 

5393 

5487 

463 

558 1 

5675 

5769 

5862 

5956 

6o5o 

6143 

6237 

633 1 

6424 

464 

65i8 

6612 

6705 

6799 

6892 

6986 

7°  79 

7*74 

7266 

7360 

465 

7453 

7546 

7640 

7733 

7826 

792c 

801 3 

8106 

8199 

8293 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9o38 

9 1 3 1 

9224 

467 

93»7 

94io 

95o3 

q5q6 

9689 

9782 

9875 

9967 

••60 

•r  53 

468 

670246 

0339 

043 ! 

o524 

0617 

0710 

0802 

0895 

0988 

ic8o 

469 

1173 

1263 

i358 

1 45 1 

1 543 

1 636 

1728 

1821 

1913 

2005 

470 

672098 

2190 

2283 

2375 

2467 

2500 

2652 

2744 

2836 

2929 

471 

3021 

3 1 13 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

385o 

472 

3942 

4o34 

4126 

4218 

43io 

4402 

4494 

4586 

4677 

4769 

473 

4861 

4g53 

5o45 

5 1 37 

5228 

53ao 

5412 

55o3 

5595 

5687 

474 

5778 

5870 

5962 

6o53 

6i45 

6236 

6328 

6419 

65ii 

6602 

475 

6694 

6785 

6876 

6068 

7o59 

ii5i 

7242 

7333 

7424 

7516 

476 

477 

mi 

2$ 

7789 

8700 

7881 

8791 

BE 

8o63 

8973 

81 54 
9064 

8245 

91 55 

8336 

9246 

8427 

9337 

478 

9428 

95i9 

9610 

9700 

979' 

9882 

9973 

••63 

•1 54 

•245 

479 

68o336 

0426 

o5i7 

0607 

0698 

0789 

0879 

0970 

1060 

ii5i 

480 

681241 

i332 

1422 

i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

481 

2145 

2235 

2326 

2416 

a5o6 

2596 

2686 

2777 

2867 

2957 

482 

3o47 

3 137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

484 

4845 

4935 

5o25 

5i  14 

5204 

5 294 

5383 

5473 

5563 

5652 

485 

5742 

583 1 

?92‘ 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

486 

6636 

6726 

681 5 

6904 

6994 

7083 

7'72 

7261 

j35i 

7440 

487 

7529 

7618 

77°7 

7796 

7886 

7975 

8064 

81 53 

8242 

833 1 

488 

8420 

85o9 

8598 

8687 

8776 

8865 

8o53 

9042 

9 1 3 1 

9220 

489 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

••19 

•107 

490 

491 

69?$ 

0285 

1170 

o3i3 

1258 

0462 

1 347 

o55o 

1435 

o63g 

1 524 

0728 

1612 

0816 

1700 

0905 

1789 

0993 

1877 

492 

1965 

2o53 

2142 

2230 

23 1 8 

2406 

2494 

2583 

2671 

2759 

4.93 

2847 

2935 

3023 

3 1 1 1 

3i99 

3287 

3375 

3463 

355i 

3639 

494 

3727 

38i5 

3903 

399i 

4078 

4166 

4254 

4342 

443o 

45i7 

495 

46o5 

4693 

4781 

4868 

4956 

5o44 

5i  3 1 

5219 

53o7 

5394 

496 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

497 

6356 

6444 

653 1 

6618 

6706 

6793 

6880 

6968 

7o55 

7U2 

498 

499 

7229 

8101 

liu 

7404 

8275 

7491 

8362 

r78 

8449 

7665 

8535 

7752 

8622 

7839 

8709 

& 

8014 

8883 

5oo 

698970 

9057 

9144 

9231 

9317 

94o4 

9491 

9578 

9664 

975 1 

5oi 

9838 

9924 

••11 

••98 

•184 

•271 

•358 

•444 

•53 1 

•617 

5oa 

700704 

0790 

0877 

0963 

io5o 

n36 

1222 

1 309 

i3o5 

1482 

5o3 

1 568 

1 684 

1741 

1827 

1913 

*999 

2086 

2172 

2258 

2344 

5o4 

243i 

2517 

2603 

2689 

2775 

2861 

2047 

3o33 

3 1 19 

32o5 

5o5 

3aoi 

3377 

3463 

3549 

3635 

3721 

3807 

38g3 

3979 

4o65 

5o6 

4161 

4236 

4322 

4408 

4494 

4579 

4665 

4]5i 

4857 

4922 

507 

508 

5oo8 

5864 

5094 

5949 

5a65 

6120 

535o 

6206 

5436 

6291 

5522 

6376 

5607 

6462 

5693 

6547 

ini 

5«9 

6718 

68o3 

6888 

6974 

7o59 

7i44 

7229 

73i5 

7400 

7485 

5io 
5i  i 

707570 

8421 

7655 

85o6 

774° 

8591 

7826 

8676 

k: 

®6 

8081 

8931 

8166 

9015 

825i 

9100 

8336 

9185 

5ia 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

••33 

5i3 

7101 17 

0202 

0287 

0371 

0458 

0540 

0625 

0710 

0794 

0879 

5i4 

oo63 

1048 

1 1 32 

1217 

i3oi 

1 385 

1470 

1 554 

1689 

1723 

5i5 

1807 

1892 

1976 

2060 

2144 

2229 

a3 1 3 

a39I 

2481 

2566 

5i6 

a65o 

2734 

2818 

2902 

2986 

3070 

3 1 54 

3a38 

3323 

3407 

5,7 

3491 

3575 

365y 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

5i8 

433o 

44i4 

4497 

458i 

4665 

4749 

4833 

4916 

5ooo 

5o84 

619 

5167 

5a5i 

5335 

5418 

55oa 

5586 

5669 

5753 

5836 

5920 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

85 

85 

85 

85 

85 

85 

84 

84 

84 

84 

84 

84 


D. 


00  00  CO  CO  00  QO  00  00  00  00  00  00  00  QO  00  00  00  go  00  QO  00  QO  OOO  O '*0  O ^0  O O 'O  O ^0  ^0  O *0  'O  'O  o o o ^0  o ,o  O ■O 
O'O'O'^O  O-J  — l QOQOOOOO  00O  OvOOvCO  O O OOO  — — UU>  U>£t  5s  £k 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


9 


N. 

1 0 

1 

2 

3 

4 

5 

6 

7 

1 8 

9 

D. 

5ao 

7i6oo3 

6087 

6170 

6254 

6337 

6421 

65o4 

6588 

J 6671 

6754 

83 

5ai 

532 

6838 

7671 

6921 

7764 

7004 

1831 

7088 

7920 

£3 

7254 

8086 

7338 

8169 

742* 

8253 

! 75o4 
8336 

-758-7 

8419 

83 

83 

5a3 

85o2 

8585 

8668 

875i 

8834 

8917 

9000 

9083 

9i65 

9248 

83 

524 

933 1 

9414 

9497 

958o 

9663 

9745 

9828 

"00 

9994 

•*77 

83 

525 

7 2oi59 

0242 

o325 

0407 

0490 

o573 

o655 

o738 

0821 

0903 

*3 

5a6 

0086 

1068 

u5i 

1233 

i3i6 

1 398 

1481 

1 563 

1646 

1728 

i 82 

521 

181 1 

i893 

i975 

2o58 

2140 

2222 

23o5 

2387 

2469 

: 2552 

82 

528 

2634 

2716 

2798 

2881 

2q63 

3o45 

3i  27 

3209 

3291 

3374 

82 

529 

3456 

3538 

3620 

37o2 

8784 

3866 

3948 

4o3o 

4112 

4194 

82 

53o 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

493 1 

5oi3 

82 

53 1 

5o95 

5i76 

5258 

5340 

5422 

55o3 

5585 

566i 

5748 

583o 

82 

532 

59I2 

5q93 

6o75 

6i56 

6238 

632o 

6401 

6483 

6564 

6646 

82 

533 

6727 

6809 

6890 

6972 

7o53 

7*34 

7216 

7297 

1 7379 

7460 

81 

534 

7541 

7624 

7704 

7785 

7866 

7948 

8029 

8110 

i 8191 

8273 

81 

535 

8354 

8435 

85i6 

8597 

8678 

8759 

8841 

8922 

9oo3 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

967° 

965 1 

9732 

981 3 

9893 

81 

537 

9974 

••55 

•i3o 

•217 

•298 

•378 

•459 

•54o 

•621 

•702 

81 

538 

730782 

o863 

0944 

1024 

iio5 

1186 

1266 

1 347 

1428 

1508 

81 

539 

1^89 

1669 

i75o 

i83o 

1911 

*99* 

2072 

2 1 52 

2233 

23 1 3 

81 

54o 

732394 

2474 

2555 

2635 

27i5 

2796 

2876 

2956 

3o37 

3i  17 

80 

54i 

3i97 

3278 

3358 

3438 

35i8 

3598 

3679 

3759 

3839 

39*9 

80 

542 

3o99 

4079 

4160 

4240 

4320 

4400 

4480 

4660 

4640 

4720 

, 80 

543 

4800 

4880 

496o 

5o4o 

5l20 

52oo 

5279 

5359 

6439 

5819 

so 

544 

5599 

6397 

5679 

5]69 

5838 

59i8 

5998 

6078 

6i5  7 

6237 

63 17 

80 

545 

6476 

6556 

6635 

67i5 

6795 

6874 

6954 

7o34 

71 13 

80 

546 

547 

7io3 

7987 

Z3 

7352 

8146 

743i 

8225 

75ii 

83o5 

££ 

& 

£8 

& 

7908 

8701 

79 

79 

548 

8781 

8860 

8939 

9018 

9°97 

9*77 

9256 

9335 

94*4 

9493 

79 

549 

9672 

9651 

9731 

9810 

9889 

9968 

••47 

•126 

•2o5 

•284 

79 

55o 

74o363 

0442 

0521 

0600 

0678 

0757 

o836 

o9i5 

0994 

io73 

79 

55 1 

1 1 52 

I23o 

1 309 

1 388 

1407 

1646 

1624 

i7o3 

1782 

i860 

79 

79 

55a 

1989 

2018 

2006 

2i75 

2254 

2332 

2411 

2489 

2568 

2647 

553 

2726 

2804 

2882 

2961 

3o39 

3i  18 

3i96 

3275 

3353 

343i 

78 

554 

35io 

3588 

3667 

3745 

3823 

39o2 

3980 

4o58 

4i36 

42 1 5 

78 

555 

4293 

437i 

4449 

4528 

4606 

4684 

4762 

4840 

49*9 

4997 

78 

556 

5o75 

5i53 

523i 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7*79 

7256 

7334 

78 

559 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8o33 

8110 

78 

56o 

748188 

8266 

8343 

8421 

8498 

8576 

8653 

873 1 

8808 

8885 

77 

56i 

8963 

9040 

9118 

9195 

9272 

935o 

9427 

9604 

9582 

9659 

77 

562 

9736 

98*4 

9«9* 

9968 

••45 

•123 

•200 

*277 

•354 

•43* 

77 

563 

7 5o5o8 

o586 

o663 

0740 

0817 

0894 

°97* 

1048 

1125 

1202 

77 

564 

1279 

1 356 

1433 

i5io 

1 58-7 

1664 

i74i 

1818 

1895 

1972 

77 

565 

2048 

2125 

2209 

2279 

2356 

2433 

25o9 

2586 

2663 

2740 

77 

566 

2816 

2893 

OT 

3o47 

3ia3 

3200 

3277 

3353 

343o 

35o6 

77 

77 

567 

3583 

366o 

3p6 

38i3 

3889 

3966 

4o4i 

4119 

4195 

4272 

568 

4348 

4425 

45oi 

4578 

4654 

473o 

4807 

4883 

4900  , 

5o36 

569 

5 1 1 2 

5i89 

5265 

5341 

5417 

5494 

557o 

5646 

5722 

5799 

76 

57o 

755875 

595i 

6°>t 

6io3 

6180 

6a56 

6332 

6408 

6484 

656o 

76 

57i 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7820 

76 

572 

573 

2% 

7548 

83o6 

7624 

8382 

£3 

2£35 

785 1 
8609 

S3 

8oo3 

8761 

Sss 

76 

574 

8912 

8988 

9063 

9*39 

9214 

9290 

9366 

944* 

95*7 

9592 

76 

575 

9668 

9743 

9819 

9894 

997° 

••45 

•121 

•196 

•272 

•347 

75 

576 

760422 

0498 

°573 

0649 

0724 

°799 

0875 

09D0 

1025 

IIOI 

75 

1176 

1028 

I2DI 

2003 

i326 

2078 

1402 

2 1 53 

*477 

2228 

1552 

23o3 

1627 

2378 

1702 

2453 

1778 

2629 

1 853 
2604 

7* 

7^ 

579 

2679 

2754 

2829 

2904 

2978 

3o53 

3 1 28 

3ao3 

3378 

3353 

75 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

LO 


A TABLE  OF  LOGARITHMS  FROM  L TO  10,000. 


N. 

0 

1 

! 2 

3 

4 

1 5 

! 

7 

8 

! 9 

b.  1 

58o 

763428 

35o3 

3578 

3653 

3727 

3802 

3877 

3952 

4027 

' 4101 

75 

58 1 

4176 

425i 

4326 

44oo 

4475 

1 455o 

4624 

4699 

4774 

1 4848 

1 75 

58a 

4q23 

499s 

5072 

5147 

5221 

5296 

i 5370 

5445 

5620 

5594 

75 

583 

566o 

5743 

58i8 

5892 

, 5966 

6041 

61 1 5 

6190 

6264 

6338 

74 

584 

6413 

6487 

656a 

6636 

67XO 

6785 

6859 

69J3 

7007 

1 7082 

-4 

*>85 

586 

7 1 56 
7808 

723o 

7972 

7304 

8046 

7379 

8120 

7453 

8194 

in 

7601 

8342 

7675 

8416 

7749 

8490 

! 7823 

1 8564 

,4 

74, 

58X 

8638 

8712 

8786 

8860 

8984 

9008 

9082 

9x56 

9230 

g3  o3 

74 1 

588 

9377 

945  J 

9525 

o5o9 

9673 

9746 

9820 

9894 

9968 

••42 

! 74 

589 

770116 

0189 

0263 

63o6 

0410 

0484 

0557 

063  X 

0706 

0778 

74 

590 

770852 

0926 

0999 

1073 

1 146 

1220 

1293 

1367 

*440 

1 5x4 

74 

5oi 

1587 

1661 

1734 

1808 

l88l 

1955 

2028 

2102 

2175 

2248 

73 

59a 

2322 

2395 

2468 

2542 

26x5 

2688 

2762 

2835 

2908 

2981 

73 

593 

3o55 

3128 

3201 

3274 

3348 

1 3421 

3494 

3567 

3640 

37x3 

73 

594 

3786 

386o 

3933 

4006 

4079 

4162 

4225 

4298 

4371 

4444 

73 

595 

4017 

4590 

4663 

4736 

4809 

4882 

4955 

5o28 

5xoo 

5173 

73 

596 

6246 

53i9 

5392 

5465 

5538 

1 56io 

5683 

5756 

5829 

5902 

73 

*97 

6974 

6047 

6120 

6193 

6265 

1 6338 

641 1 

6483 

6556 

6629 

73 

596 

6701 

6774 

6846 

6919 

6992 

I 7064 

7*37 

7209 

7282 

7354 

73 

599 

7427 

7499 

7572 

7644 

77*7 

7789 

7862 

7934 

8006 

8079 

7J 

600 

778161 

8224 

8296 

8368 

8441 

85i3 

8585 

8658 

8730 

j 8802 

72 

601 

8874 

8947 

9OI9 

9091 

9i63 

9236 

93o8 

9380 

9452 

95a4 

72 

602 

9596 

9669 

974i 

9813 

9885 

9957 

••29 

•101 

•i73 

•246 

72 

6o3 

780317 

0889 

0461 

o533 

o6o5 

0677 

0749 

0821 

0893 

0965 

73 

604 

103-7 

1109 

1181 

1253 

x 324 

1396 

1468 

1540 

1612 

1684 

1 72 

6o5 

1755 

1827 

1899 

1971 

2042 

21 14 

2186 

2258 

2329 

2401 

| 72 

606 

2473 

2544 

2616 

2688 

2759 

283 1 

2902 

2974 

3o46 

3i  17 

72 

607 

3189 

3260 

3332 

34o3 

3475 

3546 

36i8 

3689 

3761 

3832 

! 7* 

608 

3904 

3075 

4046 

4118 

4189 

| 4261 

4332 

44o3 

4475 

4546 

7* 

609 

4617 

4689 

4760 

483x 

4902 

4974 

5o45 

5i  16 

5187 

5259 

7* 

610 

785330 

54oi 

5472 

5543 

56i5 

! 5686 

5757 

6828 

6899 

5970 

7* 

61 1 

6041 

6112 

61 83 

6254 

6325 

! 6396 

6467 

6538 

6609 

6680 

7* 

61a 

6i3 

6751 

7460 

6822 

753i 

6893 

7602 

6964 

7673 

7o35 

7744 

7106 

7815 

ps 

7248 

7956 

73*9 

8027 

$8 

7* 

7* 

614 

8168 

8239 

83io 

838 1 

8451 

8522 

o593 

8663 

8734 

8804 

7* 

6i5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

7* 

616 

9581 

965i 

9722 

9792 

9863 

9933 

•••4 

••74 

•144 

*2x5 

70 

6l7 

790285 

o356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

618 

0988 

1059 

1129 

**99 

1269 

i34o 

1410 

1480 

i55o 

1620 

70 

619 

1691 

1761 

1 83 1 

1901 

*97* 

| 2041 

2 1 1 1 

2181 

2252 

2322 

70 

620 

792392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

3092 

3i6a  I 

3a3i 

33oi 

337  x 

3441 

35xi 

358i 

365i 

3721 

70 

622 

3700 

386o 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

44l8 

70 

623 

624 

4488 
5 1 85 

4558 

5254 

4627 

5324 

S3 

83 

4836 

5532 

4906 

56o2 

4976 

6672 

5045 

5741 

5 1 x 5 
58ix 

70 

7° 

1 6,5  | 

588o 

5949 

6019 

6088 

6i58 

6227 

6297  , 

6366 

6436 

65o5 

69 

626  1 

6574 

6644 

6718 

6782 

6852 

6921 

699c 

7060 

7129 

7*98  j 

69 

i M 

7268 

7337 

7406 

747* 

7545 

7614 

7683 

7752 

782* 

7890  1 

69 

J 628 

7000, 

8029 

8098 

8167 

8236 

83o5 

8374 

6443 

85i3 

8582 

69 

629 

665. 

8720 

8789 

8858 

8927 

8996 

9065 

9x34 

9203 

9272 

69 

63o 

799341 

94oq 

9478 

9547 

96x6 

9685 

9754 

9823 

9802 

9?6i 

69 

631 

632 

800029 

O7I7I 

0098 

0786 

0167 

o854 

0236 

0923 

o3o5 

0992 

0373 

1061 

0442 

1X20 

o5x  1 

1 198 

o58o 

1266 

0648 

1 335 

69 

633 

1404 

1472 

i54i 

1609 

1678 

*747 

181 5 

1884 

1952 

2021 

69 

634 

2089; 

2X58 

2226 

2296 

2363 

2432 

a5oo 

2568 

2637 

2706  | 

635 

2774 

2842 

2910 

2979  | 

3o47 

3x  16 

3x84 

3252 

3321 

3389 

68 

636 

3457 

3525 

3594 

3662 

3730 

37o8 

3867 

3935 

4oo3 

4071 

68 

637 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4016 

4685 

4753 

68 

638 

4821 

4889 

4967 

5o25 

5o93 

5i6i 

5229 

5297  ! 

5365 

5433 

68 

639 

55oi 

5569 

5637 

5705 

5773 

5841 

59o8 

5976 

6044 

6112 

68 

N. 

0 

1 

2 

3 

4 

5 i 

6 

7 1 

8 

9 

D 

A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


1J 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

640 

806180 

6248 

63i6 

6384 

645 1 

65i9 

| 6587 

6655 

6723 

6700 

68 

641 

6858 

6926 

6994 

7061 

7129 

7'97 

7264 

7332 

I40? 

7467 

68 

642 

1 643 

7535 

8211 

7603 

8270 

7670 

8346 

7738 

8414 

7806 

8481 

7878 

8540 

83 

8008 

8684 

8076 

8781 

8148 

8818 

684 

67| 

1 644 

8886 

8953 

9021 

9088 

9i56 

9228 

9290 

9358 

9425 

9492 

67 1 

645 

g56o 

9627 

9694 

9762 

9829 

9896 

9o64 

••3 1 

••9g 

•i65 

67  j 

! 646 

8io233 

o3oo 

o36-j 

0434 

o5oi 

o569 

o636 

°7°3 

0770 

0837 

67 1 

647 

0904 

0971 

io3g 

uo6 

1173 

1240 

1 307 

1874 

1441 

i5o8 

64S 

i575 

1642 

n°  9 

1776 

1843 

1910 

*977 

2044 

21 1 1 

2178 

67 

649 

2245 

2312 

2879 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

65o 

812913 

2980 

3o47 

3ii4 

3i8i 

3247 

33 1 4 

338i 

3448 

35i4 

67 

65 1 

358 1 

3648 

3714 

3781 

3848 

3oi4 

3981 

4048 

4"4 

4181 

a7 

67 

652 

4248 

43 14 

438i 

4447 

45i4 

458 1 

4647 

4714 

4780 

4847 

653 

4oi3 

4980 

5046 

5u3 

5i79 

5246 

53 1 2 

5378 

5445 

55ii 

66 

654 

5578 

5644 

5711 

5777 

5843 

59io 

5976 

6042 

6109 

6176 

66 

655 

6^41 

63o8 

6374 

644o 

65o6 

6673 

6639 

6705 

6771 

6838 

66 

656 

6904 

6970 

7086 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

65t 

658 

]565 

8226 

763i 

8292 

&2 

7764 

8424 

j83° 

8490 

85 

8028 

8688 

8094 

8754 

8160 

8820 

66 

66 

659 

8885 

8961 

9OI7 

9o83 

9149 

92l5 

9281 

9346 

9412 

9478 

66 

660 

819544 

9610 

9676 

9741 

9807 

9873 

9939 

•••4 

••70 

•i36 

66 

661 

820201 

0267 

o333 

0399 

0464 

o53o 

o5o5 

0661 

0727 

0792 

66 

662 

o858 

0924 

0080 

io55 

1120 

1 186 

I 2DI 

i3i7 

i382 

1448 

66 

663 

i5i4 

i5jg 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

2168 

2233 

2299 

2864 

2480 

2495 

2060 

2626 

2691 

2756 

65 

665 

2822 

2887 

2982 

3oi8 

3o83 

3i48 

32 13 

3279 

3344 

3409 

65 

666 

3474 

3539 

36o5 

3670 

3735 

38oo 

3865 

393o 

3996 

4061 

65 

667 

4126 

4191 

4256 

4321 

4386 

445 1 

45i6 

458i 

4646 

4711 

65 

668 

4776 

4841 

4906 

497 1 

5o36 

5ioi 

5i66 

523 1 

5296 

536i 

65 

669 

5426 

5491 

5556 

5621 

5686 

5751 

58;  5 

588o 

5945 

6010 

65 

670 

826075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65  i 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7ni 

7175 

7240 

73o5 

65 

672 

7369 

7434 

7499 

7d63 

7628 

7757 

7821 

7886 

795' 

65 

673 

8oi5 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

853i 

8695 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9a39 

64 

675 

9304 

9368 

9432 

9497 

9561 

9625 

96f 

97  54 

9818 

9882 

64 

676 

9947 

••11 

••75 

•i39 

•204 

•268 

•332 

•3o6 

•460 

•525 

64 

$ 

83o589 

i-23o 

o653 

1294 

0717 

i358 

0781 

1422 

0845 

i486 

38 

o973 

1614 

1087 

1678 

1102 

1742 

1 166 
1806 

64  i 
6*. 

679 

1870 

1984 

1998 

2062 

2126 

2189 

2253 

2317 

238i 

2445 

64  i 

680 

8325o9 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3o§3 

6 A 

681 

3i47 

3211 

3275 

3338 

3402 

3466 

3540 

35o3 

3657 

3721 

64 

682 

3784 

3848 

39I2 

3975 

4o39 

41  o3 

4166 

4280 

4294 

4357 

64 

683 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

5o56 

5l20 

5i83 

5247 

53 10 

5373 

5437 

55oo 

5564 

5627 

63 

685 

569 1 

5754 

5817 

588i 

5944 

6007 

607; 

6i34 

6197 

6261 

I 63 

I 686 

6324 

6387 

645i 

65i4 

6577 

6641 

6704 

6767 

6880 

6894 

1 63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

688 

7688 

7662 

7Il5 

77-18 

I841 

7904 

7967 

8080 

8o93 

81 56  , 

63 

689 

8219 

8282 

8345 

8408 

8471 

8534 

8597  1 

8660 

8723 

8786 

1 63 

1 690 

838840 

8912 

8975 

9o38 

9101 

9164 

922] 

9280 

9352 

941 5 

63 

691 

9473 

9541 

9604 

9667 

9729 

9792 

9855  | 

9918 

9981  ; 

••43 

63 

692 

840106 

0169 

0232 

0294 

0357 

0420 

0482  ! 

0645 

0608  1 

0671 

63 

693 

0733 

0796 

o859 

0921 

0984  j 

1046 

IlOO  | 

1172 

1234 

1297 

63 

694 

I 1 359 

1422 

1485 

; 1547 

l6lO  ; 

1672 

«735  | 

•797> 

i860 

1922 

63 

j 69s 

j i985 

2047 

2110 

; 2172 

2235 

2297 

2860  j 

2422 

2484 

2547 

62 

696 

2609 

2672 

2734 

: 2796 

2859 

:92i 

2983 

3046 

3 1 08 

3170 

62 

1 ^ 

3233 

3295 

3357 

3420 

3482  | 

3544 

36o6 

3669 

3j3i 

3793 

621 

698 

3855 

3oi8 

| 398o 

4042 

4104  ! 

4166 

4229 

4291 

4353 

44i  5 

6t  * 

699 

4477 

( 4W9 

1 4601 

4664 

4726  1 

4788 

485o  j 

4912 

4974 

5o36 

69  ! 

N. 

t~ 

L ■ _ 

2 

! 3 

._U 

LLll  I 

j 

_ 8 [ 9 

1)  | 

— J 

L2 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

}>. 

700 

845098 

5 1 60 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

701 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6i5i 

62 1 3 

6275 

62 

702 

633  7 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

703 

6955 

7017 

7079 

7 1 4i 

7202 

7264 

7326 

7388 

7449 

?5i  1 

62 

$ 

7S73 

8189 

7634 

8261 

”758 

8374 

& 

I88' 

8497 

J?5g 

8004 

8620 

8066 

8682 

, 3128 
! 3743 

62 

62 

706 

88o5 

8866 

8028 

8989 

9o5i 

9112 

9*74 

9235 

9297 

1 93  58 

61 

7°7 

9419 

9481 

9542 

9604 

9666 

9726 

9788 

9849 

99*  * 

997- 

61 

708 

85oo33 

0095 

01 56 

0217 

0279 

o34o 

0401 

0462 

0624 

o585 

61 

7°9 

0646 

0707 

0769 

o83o 

0891 

0952 

1014 

1075 

1 1 36 

**97 

61 

710 

71 1 

851258 

1870 

1320 

io3i 

i38i 

*992 

1442 

2o53 

i5o3 

2114 

1 564 

2175 

1625 

2236 

1686 

2297 

; m 

1809 

2419 

61 

61 

7ia 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

7.3 

3090 

3i5o 

3211 

3272 

3333 

3394 

3455 

3d  1 6 

3577 

3637 

61 1 

714 

3698 

3759 

3820 

388 1 

3o4i 

4002 

4o63 

4124 

4i85 

4245 

61 

7.5 

43o6 

4367 

4428 

4488 

4549 

4610 

4670 

473i 

4792 

4852 

61 

716 

4913 

4974 

5o34 

5og5 

5 1 56 

52i6 

5277 

5337 

5398 

5459 

61 

717 

55i9 

558o 

5640 

5701 

5761 

5822 

5882 

5943 

6oo3 

6064 

61 

718 

6124 

6i85 

6245 

63  06 

6366 

6427 

6487 

6548 

6608 

6668 

60 

7*9 

6729 

6789 

685o 

6910 

6970 

703 1 

7091 

7 1 52 

7212 

7272 

60 

720 

721 

857332 

7935 

7393 

7995 

7453 

8o56 

75i3 
81 16 

7^74 

8176 

7634 

8236 

7694 

8297 

$2 

7815 

8417 

7875 

»477 

60 

60 

722 

8 53 7 

8597 

865-, 

8718 

8778 

8838 

8898 

8o58 

9018 

9078 

60 

723 

724 

723 

726 

91 3y 

86o33$ 

°937 

9198 

0006 

9258 

9859 

0458 

io56 

93 1 8 

Sis 

1116 

9379 

9978 

0078 

1176 

1773 

9439 

••38 

0637 

1236 

2$ 

0697 

1295 

SS 

9610 

•218 

0817 

i4i5 

9670 

•278 

0877 

1475 

60 

60 

60 

60 

727 

1 534 

1594 

1 654 

1714 

1 833 

i8g3 

io52 

2012 

2072 

60 

728 

2 13 1 

2191 

225l 

23l0 

2370 

243o 

2489 

2549 

2608 

2668 

60 

729 

2728 

2787 

2847 

2906 

2966 

3o25 

3o85 

| 3 144 

3204 

3263 

60 

73o 

863323 

3382 

3442 

35oi 

356 1 

3620 

368o 

3739 

3799 

3858 

59 

73i 

3917 

3977 

4o36 

4096 

41 55 

4214 

4274 

4333 

4392 

4452 

59 

732 

401 1 

457o 

463o 

4689 

4748 

4808 

4867 

4926 

4985 

5o45 

59 

733 

5io4 

5 1 63 

5222 

6282 

5341 

5400 

5459 

5D19 

5578 

5637 

59 

734 

5696 

5755 

58i4 

5874 

5933 

5on2 

6o5i 

6110 

6169 

6228 

59 

735 

6287 

6346 

64o5 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

6878 

6937 

699^ 

7o55 

71U 

7173 

7232 

7291 

735o 

7400 

*9 

73^ 

7467 

8o56 

7026 
81 1 5 

7 585 

8174 

7644 

8233 

77°3 

8292 

7762 
83  5o 

7821 

8409 

7880 

8468 

ffi 

59 

739 

8644 

87o3 

8762 

8821 

8879 

8938 

8997 

9o56 

9ii4 

9*73 

59 

740 

869232 

9290 

934q 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

59 

741 

9818 

9877 

9935 

9994 

••53 

•in 

•*]? 

•228 

•287 

•345 

h 

742 

870404 

0462 

0521 

0579 

o638 

0696 

0755 

o8i3 

0872 

0930 

58 

743 

0089 

1047 

1 106 

1164 

1223 

1281 

1 339 

1 398 

1456 

i5i5 

58 

744 

1573 

1 63 1 

169O 

1748 

1806 

1 865 

I?2f 

1981 

2040 

2098 

58 

745 

2i56 

22 1 5 

2273 

233 1 

2389 

2448 

25o6 

2564 

2622  | 

2681 

58 

746 

2739 

2797 

2855 

2913 

2972 

3o3o 

3o8S  | 

3i46 

3204 

3262 

58 

$ 

3321 

3902 

M,, 

3o6o 

343 7 

40l8 

3495 

4076 

3553 

4 1 34 

36 1 1 

4192 

3669 

4250 

2720 

43o8 

3785 

4366 

3844 

4424 

58 

58 

749 

4482 

454o 

4598 

4656 

4714 

4772 

483o 

4888 

4945 

5oo3 

58 

i5o 

875061 

5i  19 

5177 

5235 

5293 

535i 

5409 

5466 

5524 

5582 

58 

761 

564o 

5698 

6756 

58i3 

5871 

5929 

5987 

6045 

6102 

6160 

58 

6218 

6276 

6333 

6391 

6449 

6607 

6564 

6622 

6680  j 

6737 

58 

753 

6795 

6853 

6910 

6968 

7020 

7083 

7141 

7*99 

7256 

73*4 

58 

7&4 

755 

7371 

7947 

7429 

8004 

7487 

8062 

7644 

81 19 

7602 

8177 

77*7 

8292 

IV4 

S' 

& 

58 

l1 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8?8*  1 

90  }9 

76-7 

9096 

9153 

9211 

9268 

9)25 

9383 

9440 

9497 

9555 1 

9612 

?7 

7^ 

9669 1 

9726 

9784 

9841 

9898 

9?56 

••i3 

••70 

•127 

•1 85 

?7 

769 

880242 

0299 

o3  56 

o4i3 

0471 

0028 

o585 

0642 

0699 

0756 

^>7 

N. 

0 1 

1 

2 

3 

4 

5 

6 

7 

8 

_?1 

D. 

A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


18 


N. 

0 

I 

2 

3 

nr 

1 5 

6 

7 

8 

9 

D. 

760 

880814 

0871 

0928 

0985 

1042 

1099 

1 1 56 

1 2 1 3 

1271 

i328 

5? 

76i 

i385 

1442 

U99 

1 556 

i6i3 

1670 

1727 

1784 

1841 

1898 

57 

T6^ 

io55 

2012 

2069 

2126 

2i83 

2240 

2297 

2354 

2411 

2468 

57 

763 

2525 

2531 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3548 

3o3i 

36o5 

57 

764 

309^ 

3i5o 

3207 

3264 

3321 

3377 

3434 

3401 

5? 

765 

366 1 

3718 

3775 

3832 

3888 

3945 

4002 

4009 

41 1 5 

4172 

57 

766 

I 4229 

4285 

4342 

4455 

4312 

4669 

4625 

4682 

88 

! 57i 

$ 

ii 

4852 

4909 

5o2  2 

5078 

5i35 

6192 

5248 

57 1 

5418 

5474 

553i 

5587 

5644 

570c 

5767 

58i3 

5870 

57 

769 

[ 5926 

5983 

6089 

6096 

6i52 

6209 

6266 

6321 

6378 

< 6434 

56 

770 

886491 

6547 

6604 

6660 

6716 

6829 

6885 

6942 

6998 

56 

77* 

7054 

7*1* 

7167 

i*£ 

7280 

•7392 

7449 

i5o5 

756’ 

56 

11 1 
773 

76i7 

8179 

7674 

18286 

273° 

8292 

8853 

itt 

7842 

8404 

7898 

8460 

lit 

801 1 

8573 

8067 

8629 

8123 

8685 

56 

56 

774 

8741 

87 97 

8909 

896  5 
9026 

9021 

9°77 

9134 

9*9° 

gj5° 

9246 

56 

775 

9802 

9358 

9414 

947° 

••3o 

9582 

9638 

9694 

9806 

56 

776 

9862 

99>8 

••86 

•14* 

**97 

•253 

•3o9 

•365 

56 

$ 

890421 

0477 

io35 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

0980 

1091 

1147 

i2o3 

1259 

1J14 

1370 

1426 

1482 

56 

779 

1537 

i593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

2i5o 

2206 

2262 

2317 

2373 

2429 

2484 

254o 

2595 

56 

781 

2707 

2762 

2818 

2873 

2929 

2985 

3o4o 

3096 

3i5i 

56 

3207 

3262 

33 1 8 

3373 

3429 

3484 

3540 

35o5 

365 1 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4538 

4039 

4094 

4i5o 

42o5 

4261 

55 

784 

43 1 6 

4371 

4427 

4482 

4593 

4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

4980 

5533 

5o36 

5091 

5146 

52oi 

5257 

53i2 

5367 

55 

786 

5423 

5478 

5588 

5644 

56q9 

5754 

5809 

5864 

5920 

55 

& 

6$ 

6o3o 

608  5 

6140 

6196 

6201 

63o6 

636i 

6416 

647* 

55 

658 1 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7i32 

7187 

7242 

7297 

7352 

7407 

7462 

l5ii 

7572 

55 

790 

79* 

897627 

8l76 

7682 

8231 

&Z 

2K 

1% 

83 

8012 
856 1 

8067 
861 5 

8122 

8670 

55 

55 

792 

8725 

8780 

8835 

8800 

8944 

8999 

9054 

9I09 

9164 

9218 

55 

793 

9273 

9828 

9383 

9437 

9492 

••39 

9547 

9602 

9656 

97*  * 

9766 

55 

794 

9821 

9875 

9930 

9985 

••94 

•149 

•203 

•258 

•3l2 

55 

795 

900367 

0422 

0476 

o53i 

o586 

0640 

0696 

0749 

1295 

0804 

0859 

55 

796 

09l3 

0968 

i5i3 

1022 

1077 

1 1 3 1 

1 186 

1240 

1349 

1404 

55 

797 

1458 

i56"] 

1622 

1676 

173 1 

1785 

1840 

1894 

2438 

1948 

54 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2I29 

2384 

2492 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

287I 

2927 

2981 

3o36 

54 

800 

903090 

3 1 44 

3 199 

3253 

3307 

336i 

3416 

3470 

3524 

3578 

54 

Soi 

3633 

3687 

3741 

3795 

3849 

39o4 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

43? 1 

4445 

4499 

4553 

4607 

4661 

54 

3o3 

4716 

4770 

4824 

4878 

49J2 

4986 

5o4o 

5094 

5i4« 

5202 

54 

804 

5256 

53io 

5364 

5418 

5472 

5526 

558o 

5634 

5688 

5742 

54 1 

8o5 

570* 

6335 

585o 

5904 

5958 

6011 

6066 

61 19 

6658 

6173 

6227 

6281 

54 

806 

638o 

6443 

6497 

655i 

66o4 

6712 

6766 

6820 

54 

807 

808 

809 

6874 

74ii 

7949 

P 

8002 

6981 

h 

7o35 

2573 

8110 

P 

8i63 

7*43 

7680 

8217 

7 *96 
7734 
8270 

725o 

£2 

73o4 
784 1 
8378 

7358 

lit 

54 

54 

54 

810 

908485 

8539 

8592 

8646 

28 

8753 

8807 

8860 

8914 

Xl 

54 

811 

9021 

9074 

9128 

9181 

9342 

9396 

9449 

54 

812 

9556 1 9610 

9663 

9716 

9770 

982I 

9877 

9930 

9984 

••37 

53 

8i3 

9 1 009 1 

0144 

0197 

025i 

o3o4 

o358 

041 1 

0464 

o5i8 

057I 

53 

814 

0624  0678 

07  i 1 

0784 

o838 

0891 

0944 

0998 

io5i 

I 104 

53 

81 5 

1 1 58 

1690I 

121 1 

1264 

1I17 

>37 

1424 

>477 

i53o 

1 584 

1637 

53 

816 

1743 

us 

i85o 

1903 

1966 

2009 

2o63  1 

2116 

2169 

53 

2223 1 

2275 

238i 

2435 

2488 

2541 

2694  | 

2647 

2700 

53 

816 

2753  2806 

2859 

2913 

2966 

3019 

3072 

3 1 25 

3178 

323 1 

53 

819 

3284 

3337 

3390 

3443 

3496 

3549 

36o2 

3655 

3708 

376i 

53 

N. 

° 1 

1 

2 

3 1 

4 

5 

6 

7 

8 

9 

D. 

L4 


A TABLE  OF  LOGARITHMS  FROM  L TO  10,000 


N. 

0 

1 • 

1 3 

3 

4 

1 5 

6 

7 

8 

1 9 

8ao 

9 1 38 1 4 

' 3867 

3920 

3973 

4026 

I 4079 

4 1 32 

4184 

4237 

4290 

821 

434' 

4396 

4449 

45oa 

4555 

! 4608 

4660 

47i3 

4766 

4819 

822 

4872 

1 4925 

4977 

5o3o 

5o83 

! 5 1 36 

6189 

5241 

5294 

1 5347 

823 

1 54oo 

5453 

55o5 

5558 

56i  1 

I 5664 

5716 

5769 

5822 

1 5875 

824 

5927 

5oSo 

6o33 

6o85 

6i38 

6191 

6243 

i 6296 

6349 

6401 

825 

l 6454 

6007 

, 655g 

6612 

6664 

67*7 

6770 

6822 

1 6875 

6927 

826 

6980 

, 7033 

■ 7085 

7 1 38 

7190 

! 7243 

7295 

7348 

74oo 

7453 

«27 

828 

| 75o6 
8o3o 

7558 

8o83 

761 1 

! 81 35 

7663 

8188 

77*6 

8240 

| 7768 
i 8293 

7820 

8345 

7873 

8397 

S3 

B£ 

829 

| 8555 

8607 

j 8669 

8712 

8764 

8816 

8869 

8921 

3973 

9026 

83o 

1919078  9130 

9183 

9 a35 

! 9281 

I 9340 

9392 

9444 

9496 

9549 

83 1 

9601 

i 9653 

9706 

9758 

, 9810 

9862 

9914 

9967 

••19 

••7, 

832 

920123 

0176 

0228 

0280 

o332 

o384 

0436 

0489 

o54i 

| o593 

833 

0645 

0697 

0749 

0801 

! o853 

0906 

0968 

1010 

1062 

- * 14 

834 

1 166 

1218 

1270 

1 3 2 2 

1 374 

1426 

1478 

i53o 

1 58a 

1634 

835 

1686 

i738 

1790 

1842 

1894 

1946 

1998 

2o5o 

2102 

2154 

836 

2206 

2258 

a3io 

2362 

24«4 

2466 

2518 

257o 

2622 

2674 

83  7 

2725 

2777 

2829 

2881 

2933 

2985 

3o37 

3089 

3 140 

3192 

838 

3244 

3296 

3348 

3399 

345 1 

35o3 

3555 

3607 

3658 

37io 

839 

3762 

38 1 4 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

924279 

433 1 

4383 

*434 

4486 

4538 

4589 

4641 

4693 

4744 

841 

4796 

4848 

4899 

495 1 

5oo3 

5o54 

5ioo 

5 1 57 

5209 

5261 

842 

53 1 2 

5364 

5415 

5467 

55i8 

55to 

5621 

5673 

! 5725 

5776 

843 

5828 

5879 

593i 

5982 

6o34 

6o85 

6137 

6188 

! 6240 

6291 

844 

6342 

6394 

6445 

6497 

6548 

6600 

665 1 

6702 

1 6754 

68o5 

845 

6857 

6908 

6q5q 

701 1 

7062 

7ii4 

7165 

7216 

7268 

1 73*9 

846 

847 

$3 

7422 

7935 

7473 

7986 

7024 

8o37 

7576 

8088 

7627 

8140 

7678 

9191 

773° 

8242 

77Sl 

8293 

1 783a 
8345 

848 

8396 

8447 

8498 

8549 

8601 

8652 

87o3 

8754 

88o5 

8857 

849 

8908 

8959 

9010 

9061 

9112 

9i63 

9215 

9266 

93i7 

| 9368 

850 

85 1 

929419 

9930 

1 9470 
9981 

9521 

••32 

2$ 

9623 

•i34 

2$ 

9725 

•236 

9776 

•287 

9827 

•338 

?4 

85a 

93o44o 

0491 

0642 

0692 

0643 

0694 

0745 

0796 

0847 

0898 

853 

0949 

1000 

io5i 

1102 

1 1 53 

1204 

1254 

i3o5 

1 356 

1407 

854 

1458 

1 609 

i56o 

l6lO 

1661 

1712 

1763 

1814 

1 865 

1915 

855 

1966 

2017 

2068 

2Il8 

2169 

2220 

2271 

2322 

2372 

2423 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

857 

2981 

3o3i 

3o8a 

3 1 33 

3 1 83 

3234 

3285 

3335 

3386 

3437 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3392 

3943 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

860 

934498 

4549 

4599 

465o 

4700 

475i 

4801 

4852 

4902 

4953 

861 

5oo3 

5o54 

5io4 

5i54 

52o5 

5255 

53o6 

5356 

5406 

5457 

862 

5507 

5558 

56o8 

5658 

5709 

5759 

5809 

586o 

5910 

5960 

863 

601 1 

6061 

61 1 1 

6162 

6212 

6262 

63i3 

6363 

6413 

6463 

864 

65 1 4 

6564 

6614 

6665 

67i5 

6765 

68i5 

6865 

6916 

6966 

86: 

7066 

7**7 

7*67 

72*7 

7267 

7317 

7367 

7418 

7468 

860 

86; 

4oi8 

8019 

7568 

8069 

7618 

8119 

7660 

8169 

77*8 

8219 

& 

7869 

8370 

7919 

8420 

7969 

8470 

86'* 

8520 

857o 

8620 

8670 

8720 

8770 

8820 

88jo 

8920 

8970 

869 

9020 

9070 

9120 

9*7° 

9220 

9270 

9320 

9369 

9419 

9469 

870 

87. 

939619' 

940018 

tl 

tSi 

97*9 

0218 

9769 

0267 

98*9 

0317 

9869 

o367 

9918 

0417 

9968 

0467 

872 

o5i6 

o566 

0616 

0666 

0716 

0765 

081 5 

o865 

091 5 

0964 

873 

!0I4 

1064 

1 1 14 

: 1 63 

’ 2 1 3 

ia63 

1 3 1 3 

1 362 

1412 

1462 

8-4 

1 5i  1 1 

i56i 

161 1 

660 

1710 

1760 

1809 

i859 

*9°9 

!958 

875 

2008. 

ao58 

2107 

21 57 

2207 

2256 

23o6 

2355 

24o5 

2455 

876 

a5o4 

a554 

2603 

2653 

2702 

2752 

2801 

285i 

2901 

2950 

877 

3ooo 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

878 

3495‘ 

3544 

3593 

3643 

369a 

3742 

3791 

384i 

389o 

3939 

879 

3989 

4038 

4088 

4i37 

4186 

4236 

4285 

4335 

4384 

4433 

N.  I 

1 

0 

1 ! 

2 

3 

4 

5 

6 

7 

8 ! 

9 

D. 


53 
I 53 
53 
53 
53 
i J3, 
53 
52 
5a! 

I 5a  I 

5a 

| 5a, 
i 5a 
, 5a  | 
| 5a1 
! 5a 1 

! ? | 

I 5*j 

i ?!: 

?! 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5i 

5o 
5o| 
5a ; 
5o 
5o ! 
5o! 
5o 
5o| 
5o 
5o 

5o 

5o 

5o 

5o 

5o 

5o 

5o 

g 

£ 

59 


D. 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


15 


N. 

0 

I 

I 2 

3 

4 

1 5 

6 

: 7 

8 

9 

D.  | 

88o 

944483 

4532 

I 4681 

463 1 

4680 

4729 

4779 

4828 

4877 

4927 

49 

88  i 

4976 

5o25 

! 5074 

5i24 

5173 

5222 

5272 

5321 

5370 

5419 

49 

88a 

5409 

55i8 

j 5567 

56i6 

5665 

5715 

5764 

58i3 

5862 

5912 

49 

883 

596i 

601c 

6o59 

6108 

6157 

6207 

6256 

63o5 

6354 

6403 

49 

884 

6452 

65oi 

! 655i 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

i 49 

885 

1 6943 

6992 

1 7°4i 

7000 

7140 

7189 

7238 

7287 

7336 

7385 

49 

m 

7434 

7483 

7532 

758 1 

763o 

7679 

7728 

7777 

7826 

7875 

49 

88i 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

83 1 5 

8364 

49 

838 

84i3 

8462 

85i  1 

856o 

8609 

8657 

8706 

8755 

8804 

6853 

49 

889 

8902 

895 ; 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

j 49 

890 

949390 

9439 

9488 

953» 

j585 

9634 

9683 

973i 

9780 

9829 

! 49 

891 

_9M 

9926 

9975 

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•121 

•170 

•219 

•267 

•3 1 6 

1 49 

892 

9Do365 

0414 

1 0462 

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o56o 

0608 

0657 

0706 

0754 

o8o3 

1 49 

893 

o85i 

0900 

1 0949 

°997 

1046 

1095 

114I 

1192 

1240 

1289 

49 

894 

i338 

1 386 

i43o 

1483 

1 532 

i58o 

1629 

1677 

1726 

1775 

49 

895 

1823 

1872 

I Q20 

1969 

2017 

2066 

2114 

2i63 

2211 

2260 

48 

896 

23o8 

2356 

24o5 

2453 

25o2 

255o 

2599 

2647 

2696 

2744 

48 

897 

2792 

2841 

28S9 

2938 

2986 

3o34 

3o83 

3 1 3 1 

3 180 

3228 

48; 

898 

3276 

3325 

3373 

3421 

3470 

35i8 

3566 

36i5 

3663 

3711 

48 

899 

3760 

38o8 

3856 

3go5 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

4291 

4339 

4387 

4436 

4484 

4532 

458o 

4628 

4677 

48 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5oi4 

5o62 

5i  10 

5i58 

48 

902 

5207 

5255 

53o3 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

903 

5688 

5736 

5784 

5832 

588o 

5920 

5976 

6024 

6072 

6120 

48 

904 

6168 

6216 

6265 

63 1 3 

636i 

6409 

6457 

65o5 

6553 

6601 

48 

9o5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

906 

7128 

7176 

7224 

7272 

7320 

7368 

74i6 

7464 

7512 

48 

907 

1 908 

7607 

8086 

7655 

8i34 

I'ES! 

7761 

8229 

7799 

8277 

llil 

7942 

8421 

23S 

8o38 

85i6 

48 

48 

9°9 

8564 

8612 

; 8659 

8707 

8755 

88o3 

885o 

8898 

8946 

8994 

48 

910 

959041 

9089 

9l37 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

48 

Oil 

9518 

9566 

! 9614 

9661 

97°9 

9757 

9804 

9802 

9900 

9947 

48 

912 

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•i85 

•233 

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•376 

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48 

913 

960471 

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0661 

0709 

0756 

0804 

o85i 

0899 

48 

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0946 

0994 

1041 

1089 

1 1 36 

1184 

I 23 1 

!279 

i326 

1374 

47 

qi5 

1421 

1460 

i5i6 

1 563 

1611 

1 658 

1706 

1753 

1801 

1848 

47 

916 

1895 

1940 

1990 

2o38 

2o85 

2l32 

2180 

2227 

2275 

232a 

47 

9 >7 

2o6? 

2417 

2464 

25l  I 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3o32 

3079 

3i26 

3 1 74 

3221 

3268 

47 

9»9 

33i6 

3363 

34io 

3457 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

3835 

3882 

3929 

3977 

4024 

4071 

41 18 

4i65 

4212 

47 

921 

4260 

4307 

4354 

44oi 

4448 

4495 

4542 

4590 

463 7 

4684 

47 

922 

473i 

4778 

4825 

4872 

4919 

4966 

5oi3 

5o6i 

5io8 

5i  55 

47 

923 

5202 

5 249 

5296 

5343 

5390 

5437 

5484 

553 1 

5578 

5625 

47 

924 

5672 

5719 

5766 

58i3 

586o 

59°7 

6954 

6001 

6048 

6095 

47 

! 9^  | 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

65i  7 

6564 

47 

1 9a6 

66ll 

6658 

6705 

6752 

6799 

6845 

6892 

69I9 

6986 

7o33 

47 

! 927  ! 

7080 

7127 

7173 

7220 

7267 

73i4 

736i 

74o8 

7454 

75oi 

47 

: 928 

1 909  i 

7548 

8016 

7595 

8062 

7642 

8109 

7688 

81 56 

7735 

8203 

7782 

8249 

7829 

8296 

2!l! 

7922 

8390 

7969 

8436 

47 

47 

93o 

968483 

OO 

CJ* 

GJ 

O 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

47 

8960 

8996 

9043 

909c  i 

9136 

9i83 

9229 

9276  | 

9323 

9069 

47 

! 932 

94i6| 

9403  j 

95oc 

9556 

9602 

9649 

9693 

9742  1 

9789 

9835 

47 

a33  ; 

9882 

9928  j 

9975 

••21 

••68 

•ii4 

•161 

•207  j 

•254  | 

•3oo 

47 

! 93^ 

97o347 

o3o3  1 

0440 

0486 

o533 

0579 

0626 

0672 

0719  j 

0765 

46 

i 

0812 

o858 

0904 

0951 

0997 

1044 

1090 

1137 

1 1 83  1 

1229 

46 

| 936 

1276 

1 3 2 2 

1369 

1 41 5 

1461 

i5o8 

1 554 

1601 

1647  ! 

i6o3 

46 

937 

1740 

1786 

1 832 

1879 

1926 

1 97 1 

2018 

2064 

2110 

2157 

46 

1 93^ 

2203 

2249 

2295 

2342 

2388 

2434  | 

2481 

2527 

2^73 

2619 

46 

939 

2666 

2712 

2758 

2804 

285 1 

2897 

2943 

2989 

3oj5 

3o82 

46 

N 

0 

' 1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

2 a 


16 


A TABLE  OF  LOGARITHMS  FROM  1 TO  10,000. 


A TABLE 


OF 

LOGARITHMIC 

SINES  AND  TANGENTS 


FOR  EVER! 

DEGREE  AND  MINUTE 

OF  THE  QUADRANT. 


Remark.  The  minutes  in  the  left-hand  column  of  eacu 
page,  increasing  downwards,  belong  to  the  degrees  at  the 
bop ; and  those  increasing  upwards,  in  the  right-hand  column, 
belong  to  the  degrees  below. 


18 


<0  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 

Sine 

1 D- 

1 Cosine 

D. 

Tan*. 

IX 

Cotang. 

0 

2 

3 

4 

5 
t 

l 

9 

10 

n 

13 

13 

14 

15 

16 

\l 

!9 

30 

21 

33 

33 

34 

35 

36 

a 

B 

31 
3s 

33 

34 

35 

36 

a 

3o 

40 

41 

43 

43 

44 

fr' 

Sj 

g 

5i  1 
53  1 
53 

*4 

55 

56 

5? 

s 

6 403726 
764*756 
940847 

7 065786 
162696 
341877 

308824 

3b68ic 

417968 

453725 

7-5o5ii8 

542906 

577668 

609853 

639816 

667845 

694173 

718997 

742477 

764754 

7-785943 

806146 

82545i 

843934 

861662 

878695 

895085 

910879 
9261 19 
940842 

7- 955o83 
968870 
982233 
995108 

8- 007787 
020021 
031919 
043601 
064781 
065776 

8 07650c 
086965  I 
097183  | 
107167  1 
1 16926  . 
126471 
i358io  | 
144953 
163907  | 
162681 

3-171280  ; 
179713 
i87985 
196102 
204070 
2ii8o5  , 
219561  1 
227134 

234557 

241856 

5017-17 

2934-86 

2o82-3i 

1 i6i5  17 
i3io-68 

1 1116-75 
966-63 
852-54 
762-63 
j 689-88 

629-81 
579.36 
536-41 
499 • 38 
467.14 
438-8i 

7|3'?2 
391 -35 

336-72 

3si.75 

3o8-o0 

&S 

273-17 

263-23 

237-33 

229- 80 
222-73 
216-08 
209-81 
206-90 
198-31 
193-02 
108-01 

1 83  - 25 
178-72 

174-41 

170-31 

166-39 

162-66 

159-08 

156-66 

15*2-36 

149-24 

146-22 

143-33 

140-54 
137-86 
136-29 
i32-8o 
i3o-4i 
128- 10 
12.5-87 

123-72 

1 2 1 64 

1 19-63 

10-000000 

000000 

000000 

000000 

000000 

000000 

1 9-999999 
999999 
999999 
, 999999 

999998 

9.999998 
999997 
999997 
999996 
999996 
999995 
999995 
999994 
999993 
999993 
9.999992 
999991 
99999° 
9999  ’9 
999988 
999988 
999987 
999986 
999985 
999983 

9.999982 

999981 

999980 

999979 

999977 

999976 

999975 

999973 

999972 

999971 

9 999969 
999968 
999966 
999964 
999963 
999961 
999950 
999958 
999956 
999954 
9.999952 

99995° 

999948 

999946 

999944 

999942 

999940 

999938 

999936 

999934 

•00 

•00 

•00 

•00 

•00 

•01 

•01 

01 

• -01 

•01 

•01 

-01 

•01 

-01 

-01 

•01 

•01 

•01 

•01 

•01 

•01 

•01 

•01 

•02 

•02 

•02 

•02 

•02 

•02 

-02 

•02 

-02 

•02 

•02 

•02 

•02 

•02 

•02 

•02 

•02 

•02 
•02 
-02  , 
•o3 
•o3 
•o3 
•o3 
•o3 
•o3 
•o3 

•o3 

•o3 

•o3 

•o3 

•o3 

•04 

•04 

•04 

.04 

•04 

0 • 000000 

6- 463726 
764756 
940847 

7- 065786 

i 162696 
; 241878 

3o8825 
' 3668i7 

I 417970 
463727 

7-5o5i2o 

542909 

g& 

639820 

667849 

694179 

719004 

742484 

764761 

7-785951 

8o6i55 

82.5460 

843944 

861674 

878708 

895099 

910894 

926144 

94o858 

7- 955ioo 
968889 
982253 
995219 

8- 007809 
020045 
o3io45 
043527 
054809 
o658oo 

8-076531 
086997 
097217 
107202  j 
1 16963 
126510 
i3585i 
144996 
153952 
162727 

8-171328 

:s& 

196156 

204126 

21 io53 
219641 
227196 
234621 
241921 

5017-17 
2934-83 
2082 -3i 
161 5 - 17 
1319-69 
1115-78 

8g:g 

762-63 

689-88 

629-81 

679-33 

536-42 

499-39 

467-15 

438-82 

4 1 3 - 73 
391 -36 
3*7 1 • 28 
35 i- 36 

336- 73 

321-76 

3o8 • 06 
2q5 • 49 
283-90 
273-18 
263-25 
254-oi 
245-40 
237.35 

229-81 
222-75 
216-10 
209-83 
203-92 
198-33 
io3.o5 
188 -o3 
i83 • 27 
178-74 

174-44 

170-34 
166-42 
162-68 
159. 10 

1 55 • 68  j 

i52-4i 

149-27 

146-27 

143-36 

140-57 

132-84 
i3o-44 
128-14  1 
125-90  j 
123-76 
121 -08 
119-67 

Infinite. 
;3- 536274 
235244 
o59i53 

12-934214 

837304 

758122 

691 175 

633 1 83 
582o3c 
536273 

1 i • 494880 
457091 
422328 
390143 
36oi8o 
332 1 5i 
3o582 1 

Sag 

235239 

12-214049 

193845 

174540 

1 56o56 
138326 
121292 

1 0490 1 
089106 
075866 
059142 

12-044900 
o3 1 1 1 1 
017747 
004781  | 
11-992191 

956473 
945191 
934i94 
11-923469 
9i3oo3  1 
002783  | 

te? 

873490 

804149 

855oo4 

846048 

837273 

11-828672  | 
8202J7  | 
811964  j 
803844 
795874 
788047 
780359 
772805 
765379 
758079 

60 

a 

13 

Is 

1 53 
52 
5i 
| 5o 

!g 

2 

45 

44 

43 

42 

4i 

4o 

a 

2 

35 

34 

1 33 

32 
! 3 1 
3o 

!g 

a 

25 

24 

23 

22 

21 

20 

S 

a 

i5 

14 

i3 

12 

11 

10 

§ 

? 

5 

4 

3 

2 

I 

0 

CoHine 

I). 

Sine  | 

Cotang. 

D.  | 

Tang. 

M. 

(89  DKGRKK8.) 


SINES  AND  TANGENTS.  (1  DEGREE.) 


M. 

Sine 

D. 

CoBino 

D. 

Tang. 

D. 

Cotang. 

0 

1 

2 

3 

4 

5 

5 

I 

9 

10 

U 

12 

13 

14 

15 

16 

\l 

*9 

20 

21 

22 

23 

U 

25 

26 

3 

II 

31 

32 

33 

34 

35 

36 

11 

3, 

40 

41 

42 

43 

44 

45 

46 

! % 

| 5o 

1 51 

1 52 

53  1 

54  i 

56 

u 

& 

8-241855 

249033 

256094 

263042 

260881 

276614 

283243 

289773 

296207 

302546 

308794 

8- 314904 
321027 
327016 
332924 
338753 
344604 
35oi8i 
355783 
36i3i5 
366777 

8-372171 

3% 

387962 

394101 

398179 

403199 

408161 

4i3o68 

4i79>9 

8-422717 
427462 
4321 56 
4368oo 

441394 

445941 

45o44o 

454893 

459301 

463665 

8-467985 

472263 

476498 

48o6o3 

484848 

488963 

493040 

497078 

5oio8o 

5o5o45 

8-608974 

612067 

516726 

52od5i 

524343 

528102 

53i828 

535523 

539186 

542819 

119-63 
117-68 
u5-8o 
113-98 
112-21 
iio-5o 
io8-83 
107-21 
io5-65 
io4- i3 
102-66 

IOI-22 

90-82 

96-47 

94-60 
93- 38 

92'*? 
91  -o3 
89-90 

88-  80 
87-72 
86-67 
85-64 
84-64 

83-66 

82-71 

79-96 

#3 

K 

75-77 

74-99 

74-22 

-f3-46 

72-73 

72-00 

71-29 

70-60 

69.91 

69-24 

68*59 

67.94 

67.31 

66-69 

66-o8 

65-48 

64-89 

64-3i 

63.76 

63-19 

62-64 

62-11 
6i-58 
61  -06 
6o-55 
60-04 

9.999934 

999932 

999929 

999927 

999925 

999922 

999920 

999918 

9999*5 

999913 

999910 

9.999907 

999905 

999002 

999899 

999897 

999894 

999801 

999888 

999885 

999882 

9.999879 

999876 

999873 

999870 

999867 

999864 

999861 

999858 

999854 

999851 

9-999848 

999844 

999841 

999838 

999834 

999831 

999827 

999823 

999820 

999816 

9.999812 

999809 

999805 

999801 

999797 

999793 

999700 

999786 

999782 

999778 

9*999774 

999769 

999765 

99976i 

999757 

999753 

999748 

999744 

999740 

999735 

•04 

• 04 
•04 

• 04 
•04 

1 -04 
■04 
-04 
.04 
-04 
•04 

.04 

•04 

• 04 

• o5 

• o5 

• o5 

• o5 

• o5 
•o5 

• o5 

•o5 

•o5 

-o5 

• o5 

• o5 
■o5 
-o5 

• o5 

• o5 

• 06 

-06 

•06 

•06 

•06 

•06 

•06 

•06 

-06 

•06 

-06 

• 06 

• 06 

• 06 

• 06 

•07 

•07 

•07 

•07 

.07 
■ 07 

.07 

•07 

.07 

•07 

.07 

.07 

•07 

•07 

•07 

•07 

| 8-241921 
249102 
256i65 
j 263 1 1 5 
269966 
276691 
283323 
289856 
296292 
302634 
3o8884 

8-3i5o46 
321122 
327114 
333025 
338856 
344610 
360289 
. 3558o5 

36i43o 
366895 

8*372292 

377622 

382889 

388oo2 

393234 

398316 

4o3338 

4o83o4 

4i32i3 

418068 

8-422869 

’427618 

4323i5 

436962 

44i56o 

446110 

460613 

455070 

459481 

463849 

8-468172 

472464 

476693 

480892 

485o6o 

489170 

493260 

497293 

5oi2o8 

505267 

8-509200 

5 1 3oo8 
516961 
520790 
524606 
528349 
532o8o 
535779 

539447 

543084 

119-67 

::££ 
114-02 
112-25 
; 110-54 
108-87 
107-26 
106-70 
io4- 18 
102-70 

101-26 

97.19 

95-90 

94-65 

93-43 

92-24 

91  -08 

89-95 

88-85 

S# 

85-70 

84-70 

83-7i 

82-76 

81.82 
80-91 
80 -02 

$2 

n5 

75. 83 
75.  o5 
74-28 
73.52 
72.70 
72-06 

71-35 
70-66 
69.98 
69-31 
68-65 
68-01 
67.38 
66.76 
66- 15 
65-55 

64.96 
64-39 
63-82 
63-26 
62-72 
62-18 
6i-65 
61 . i3 
60-62 
6o- 12 

11-758079 
750890 
743835 
736885 
l 730044 

1 723309 

' 716677 

710144 
703708 
697366 
691 1 16 

11-684954 

678878 
672886 
666975 
661 144 
655390 
6497*1 
644 1 o5 
638570 
633 1 o5 

11-627708 

622378 

617111 

611908 

606766 

6oi685 

696662 

591696 

506787 

58i932 

x 1 -577131 
572382 
567685 
563o38 
558440 
553890 
549387 
544o3o 
540619 
536i5i 

1 1 • 53i828 
527546 
5233ot 
519108 

5 14950 
5io83o 
506750 
502707 
498702 
494733 

1 1 - 490800 
406902 
483o39 
479210 
475414 
47i65i 
467920 
464221 
46o553 
456916 

60 

£ 

52 

5i 

5o 

8 

8 

45 

44 

43 

42 

41 

40 

8 

8 

35 

34 

33 

32 

3i 

3o 

3 

3 

25 

24 

23 

22 

21 

20 

3 

3 

i5 

14 

i3 

12 

11 

10 

1 

2 

5 

4 

3 

2 

1 

0 

1 Cosine 

D. 

Sine  1 

Cotang. 

D.  | Tang 

16 


(88  DRORKKB.) 


i ix  wwwwwww(*ioj  cou  u u m u u 

I o <6  ocs-J  O'  c*^  ~ o o ao~j  0'<j»Jxuj»o—  o >o  oo~j  c>ui^  cx>  »o  — o-~c  oo-j  O'  t^.ix 


20  (2  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 


Sino 


0 

1 

2 

3 

4 

5 

6 

l 

9 

10 


8*542819 

546422 

549995 

5535^9 

557054 

56o54o 


563999 
56743 1 
570836 
574214 
577566 


11 

12 

13 

14 

15 

16 

\l 

*9 

20 


8 580892 
584193 
587469 
69072 1 
593948 

597 1 52 

6oo332 

6o348q 

60662J 

609734 


21 

22 

23 


8-612823 

615891 

618937 

621962 

624965 

627948 

630911 

633854 

636776 

639080 


8-642563 
645428 
648274 
661102 
65391 1 
656702 
659475 
662230 
664968 
667689 

8-67o3o3 

673080 

675751 

678405 

681043 

683665 

686272 

688863 

691438 

693998 


8 696543 
699073 
701589 
704090 
706577 
709049 
7ii5o7 
713952 
716383 
718800 


Cornua 


D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

6004 

9-999735 

•07 

8 - 543o84 

6o- 12 

1 1 -456oi6 
453309 

1 60 

59-55 

999731 

•07 

546691 

55o208 

59-62 

5q 

59-06 

999726 

•°7 

5o- 14 

449732 

58 

58-58 

999722 

-08 

5538i7 

58-66 

446i83 

57 

58-n 

999717 

.08 

557336 

58-ig 

442664 

56 

57-65 

999713 

-08 

560828 

57-73 

430172 

55 

57-19 

999708 

•08 

564291 

57-27 

436700 

432273 

428863 

; 54 

56-74 

999704 

•08 

567727 

56-82 

1 53 

56-3o 

999699 

-08 

571137 

56-38 

52 

55-87 

999604 

999689 

•08 

574520 

55- 95 

425480 

5i 

55-44 

•08 

577877 

55-52 

422123 

5o 

55-02 

9-999686 

• 08 

8-58i2o8 

55-io 

1 I -418702 
415486 

49 

54-6o 

qqq68o 

•08 

5845 1 4 

54-68 

48 

54- 19 

999675 

•08 

587795 

54-27 

412205 

47 

53-79 

999670 

999665 

.08 

591061 

53-87 

408949 

46 

53-  3g 

.08 

594283 

53-47 

405717 

45 

53-oo 

qqq66o 

•08 

597492 

53.o8 

402608 

44 

52-6i 

999655 

•08 

600677 

6o38Jo 

606978 

52-70 

399323 

396161 

43 

52-23 

qqq65o 

.08 

52-32 

42 

5i-86 

999645 

.09 

5i  -o4 

5i  -58 

3o3o22 

4i 

5i  49 

999640 

.09 

610094 

389906 

4o 

5i  • 12 

9-999635 

•09 

8-613189 

5l  -21 

i i -3868i 1 

39 

60-76 

999629 

•°9 

616262 

5o-85 

383738 

38 

5o-4i 
5o  • oG 

999624 

999619 

999614 

• 09 

619313 

5o-5o 

aa  - a; 

380687 

3? 

49*72 

•09 

.09 

625352 

00*10 

49-81 

J77601 

374648 

35 

49-38 

999608 

.09 

628340 

49-47 

371660 

34 

40-04 

999603 

-09 

63i3o8 

4Q-  i3 

368692 

33 

48-71 

999597 

-09 

634256 

48-80 

366744 

362816 

32 

48-39 

999502 

-09 

637184 

48*48 

3i 

48-00 

999586 

•°9 

640093 

48.16 

359907 

3o 

47-75 

9- 99958 1 

.09 

8*642982 

47-84 

1 1 -357018 

29 

47-43 

47-12 

46-82 

999575 

.09 

645853 

648704 

65i537 

47-53 

354147 

351290 

348463 

28 

999570 

999564 

• 09 
.09 

47-22 

46.91 

S 

46-52 

999558 

. ij 

654352 

46-61 

345648 

26 

46-22 

999553 

•10 

657149 

46- 3i 

34285i 

24 

45-92 

999547 

*10 

659928 

46-02 

340072 

23 

45-63 

999541 

>10 

662089 

45.73 

3373ii 

22 

45-35 

999535 

•10 

665433 

45-44 

334567 

21 

45.o6 

44-79 
44-  5i 

999529 

9-999524 

999518 

•10 

• 10 

- 10 

668160 

O .a 

45-26 

33 1 840 

1 1 - 3291 3o 

326437 

20 

O'070o70 

673563 

44*oo 

44-6i 

:s 

44-24 

43-97 

43.70 

999512 

999506 

999500 

- 10 

- 10 

•10 

676239 

44.34 

44-17 

43-8o 

323761 

;j 

678900 

681644 

321 100 

3 i 8456 

16 

i5 

43*44 

999493 

999487 

•10 

684172 

43.54 

; 15828 

14 

43-i8 

• 10 

686784 

43.28 

c I 32 16 

i3 

42*92 

99948i 

•10 

689381 

43 -o3 

3lo6l9 

12 

42-67 

999475 

• 10 

691963 

42-77 

3o8o37 

11 

42-42 

999469 

•10 

694629 

42-52 

3o547i 

(0 

42-17 

9-999463 

•II 

8-697081 

42-28 

11  302919 
3oo383 

9 

41-92 

999456 

-II 

699617 

42- o3 

6 

41-68 

4i-44 

99945o 
999443  ' 

• II 

-II 

702 1 3o 
704640 

£:S  1 

297861 

296354 

z 

41  -21 

999437 

• II 

707140 

41  -32 

292860 

5 

40-97 

99943 1 

• II 

709618 

41  -08 

290382 

4 

40-74 

999424 

•II 

7 1 2o83 

4o-85 

3 

4o- 5i 

999418 

• II 

714534 

40-62 

2 

4o-2o 

9994H 

• II 

716072 

4o-4o 

283028 

1 

4o-o6 

999404 

• II 

719396 

4o- 17 

280604 

0 

I). 

Sine 

Ootaiig. 

D.  I 

Tang. 

M. 

(87  UKOKKKH.) 


I gMTSirS 


SINKS  AND  TANGENTS  (3  DEGRBKS. 


21 


M. 


3 

4 

5 

6 

l 

9 

10 


la 

i3 

U 

15 

16 

>9 

20 

21 

22 

23 

24 

25 

26 

2 

29 

30 

31 
3a 

33 

34 

35 

36 

ll 

39 

40 

41 

42 

43 

44 

45 

46 

a 

is 

5 

5a 

53 

54 

55 


Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

8-718800 

40-06 

9-999404 

•11 

8-719396 

40-17 

1 1 - 280604 

60 

721204 

39-84 

999398 

• 11 

721806 

39-90 

278194 

*9 

723596 

39-62 

999391 

-11 

724204 

39-74 

275796 

58 

725072 

728447 

730688 

39-4i 

999384 

•11 

726588 

39-32 

273412 

£ 

ip 

999378 

999371 

• 11 

• 1 1 

728059 

731317 

39 -3o 

fcffi 

271041 

268683 

! 56 

55 

733027 

38-77 

999364 

• 12 

73366$ 

266337 

54 

735354 

38-  57 

999357 

• 12 

■735996 

,08317 

38-68 

1 264004 

53 

737667 

38-36 

99935c 

• IS 

1 38-48 

261683 

1 5a 

739969 

38  16 

999343 

• 12 

! 740626 

i 3827 

I 259374 

5i 

742259 

37-96 

999336 

12 

74292s 

38-07 

257078 

5o 

8-744536 

37*76 

9.999329 

• 12 

8-746207 

37-87 

1 i • 254793 

49 

746802 

37-06 

999322 

- 12 

747479 

37-68 

252521 

48 

749055 

37-37 

9993 1 5 

• 12 

749740 

37-49 

2 5o26o 

47 

751297 

753528 

755747 

757955 

37' 17 

999308 

• 12 

751989 

37-29 

24801 I 

46 

36-98 

36-79 

36-6i 

999301 

999204 

999286 

-12 

• 12 

• 12 

754227 

756453 

758668 

37-10 

36-92 

36-73 

245773 

243547 

241332 

45 

44 

43 

7601 5 1 

36-42 

999279 

• 12 

760872 

36-55 

239128 

42 

762337 

36-24 

999272 

• 12 

763o65 

36-36 

236935 

4i 

764511 

36- 06 

999265 

• 12 

766246 

36- 18 

234754 

4o 

8-766675 

35-88 

9.999257 

• 12 

8-767417 

36-00 

1 1-232583 

39 

768828 

35-70 

999250 

• i3 

769678 

35-83 

230422 

38 

770970 

773101 

35-53 

35-35 

999242 

999235 

• i3 

• i3 

7717*7 

773866 

35-65 

35-48 

228273 

226134 

IS 

775223 

35-i8 

999227 

• i3 

775995 

35- 3i 

224005 

35 

777333 

35-oi 

999220 

• i3 

778114 

35-i4 

221886 

34 

779434 

34-84 

999212 

• i3 

780222 

34-97 

219778 

217680 

216592 

33 

i8i524 

34-67 

999205 

• i3 

782320 

34-  80 

32 

7836o5 

34  - 5i 

999107 

999189 

- 13 

784408 

34*64 

3i 

785675 

34-3i  * 

• i3 

786486 

34-47 

2 1 35 1 4 

3o 

8-787736 

34-i8 

9-999181 

• i3 

8-788554 

34  • 3 1 

11 -211446 

*9 

7«9 787 
791828 

34-02 

33-86 

999 1 74 
999 1 66 

• i3 

• i3 

790613 

792662 

34- 15 
33-99 

209387 

207338 

28 

27 

793859 

33-70 

999168 

• i3 

794701 

33-83 

205299 

26 

25 

795881 

33.54 

999 1 5o 

• i3 

796731 

33-68 

203260 

797894 

33  39 

999142 

• i3 

798752 

800763 

802765 

33-52 

201248 

24 

799897 

801892 

33-23 
33- 08 

999134 

999126 

• i3 

• i3 

33-37 

33-22 

199237 

197235 

196242 

23 

22 

803876 

32-93 

9991 18 

• i3 

804768 

33-07 

21 

8o5852 

32-78 

999 1 1 0 

• i3 

806742 

32-92 

193268 

20 

8-807819 

3a-63 

9-999102 

• i3 

8-808717 

32-78 

in • 191283 

19 

809777 

32-49 

999094 

•14 

8io683  1 

32-62 

189317 

l8 

811726 

32-34 

999086 

•14 

812641 

32 -4& 

187369 

18541 1 

17 

81 3667 

3a- 19 

999077 

•14 

814689 

32-33 

l6 

815599 

32-o5 

999069 

14 

8i6529 

32-19 

183471 

i5 

817522 

3 1 -91 

999061 

•14 

818461 

32- o5 

i8i53o 

14 

819436 

3i-77 

999053 

•14 

820384 

3i  -91 

179616 

i3 

821343 

3* -63 

999044 

• 14 

822298 

177702 

12 

823240 

1 3i -4q 

999036 

14 

824205 

176795 

11 

825i3o 

3i  -35 

999027 

• 14 

826103 

3 1 -5o 

17389] 

10 

8-82701 1 

3 1 22 

9-999019 

•14 

8-827992 

829874 

3 1 - 36 

1 1 -172008 

9 

828884 

3i-o8 

9990 1 0 

• 14 

3 1 - 23 

170126 

8 

830749 

30-96 

999002 

•14 

831748 

3i  • 10 

168252 

7 

832607 

3o-82 

998998 

• 14 

8336 1 3 

30-96 

166387 

6 

834456 

30-69 

998984 

•14 

835471 

3o-83 

164629 

5 

836297 

3o-56 

998976 

•14 

837321 

30-70 

ii 62679 
160887 

4 

838i3o 

3o-43 

s&s 

• i5 

839163 

30-67 

3 

839966 

3o-3o 

• i5 

840998 

842825 

3o-45 

159002 

2 

841774 

3o- 17 

998960 

• i5 

3o-3a 

167175 

1 

843585 

3o-oo 

998941 

• i5 

844644 

3o- 19 

155356 

0 

CoBine 

D.  ! 

Sine 

Cotang.  1 

D. 

Tang. 

M. 

(86  DKORKK8.1 


22 


(4  DEGREES.)  A TABLE  OF  LOGARITHM  I C 


M. 

Sine 

D. 

Coeino 

D. 

Tang. 

D. 

Cotang. 

1- 

0 

2 

3 

t 

i 6 

1 * 

! ,1 

u 

12 

13 

14 

15 

16 

■9 

2° 

I 31 

i 22 
! 23 

I 24 

s 

3 

III 

31 

32 

33 

34 

35 

36 

II 

3, 

40 

41 

42 

43 

44 

45 

46 

IS 

1 g 

! 5i 

52 

53 

54 

55 

56 

ll 

8- 843585 
845387 
847183 
84897 1 
860701 
852525 
854291 
856049 
857801 
859546 
86n83 

8-863oi4 
864738 
866455 
868 1 65 
869868 

8t 1 565 
873255 
874938 
876615 
878285 
8-879949 

881607 

883258 

884903 

886542 

888174 

889801 

891421 

893o35 

894643 

8-896246 

897842 

899432 

901017 

902596 

904169 

905736 

S&Z 

910404 

8-911949 

913488 

9l5o22 

9i655o 

918073 

919591 

921103 

92s5io 

9*4112 

926609 

8 927100 
92«58t 
930068 
93 1 544 
933oi 5 
934481 
935942 
937)08 
938850 
940296 

3o-o5 
29-02 
29- 80 

29-55 
29  43 
29'3i 
29- 19 

£2 

28-84 

28-73 
2861 
28  -5o 
28-39 
28-28 
28-17 
28-06 

27-73 

27-63 

27-52 

27-42 

27-31 

27-21 

27-11 

27-00 

26-90 

26-80 

26-70 

26-60 

26-5i 

26-41 

26-3i 
26-22 
26-12 
26 -o3 
25-93 
25-84 
25-75 

25-66 

25- 56 

26- 47 
25-38 
25-29 
25-20 
25-12 
25  o3 
24-04 
24-86 

24-77 

24-69 

24-60 

24-52 

24-43 

24-35 

24-27 

2419 

24-11 

24  o3 

9 •99894i 
998932 
998923 
998914 
998905 
998806 
998887 

$gg 

998860 

998851 

9-998841 

998832 

998823 

998813 

998804 

998705 

998785 

998776 

998766 

998767 

9.998747 

998738 

998728 

998718 

998708 

998609 

998689 

998679 

998669 

998669 

9-998649 
998639 
998629 
998619 
998609 
998599 
998589 
99857 8 
998568 
998558 

9 • 998548 
998537 
998527 
998516 
998606 
998405 
998485 
998474 
998464 
998453 

9-998442 
99843 1 
998421 
998410 
998399 
998388 
998377 
998366 
998355 

998344 

• i5 

• i5 

• i5 

1 ,5 

• i5 

:j| 

i 5 

■ Is 

• i5 

• i5 

• i5 

• i5 

• 16 

• 16 

"a 

1 'a 

■ 16 

•16 

• 16 

• 16 

• 16 
•16 
•16 
•16 
- 16 

• 16 

• 16 
- 16 

•n 

•17 

•17 

•17 

-17 

•17 

•17 

•17 

•17 

•17 

• *7 
•17 

•17 

•*7 

::z 

• 18 

• 18 

• 18 

• 18 
-18 

18 

• 18 

• 18 

• 18 

• 18 
.18 
.18 

• 18 

• 18 

• 18 

• 18 

8-844644 
846455 
848260 
85oo57 
85 1846 
853628 
8554o3 
857171 
8589)2 
860686 
862433 

8-864173 

865906 

867632 

869351 

871064 

872770 

874469 

876162 

877849 

879529 

8 -881202 
882869 
88453o 
886 1 85 
887833 

889476 
891 1 12 
892742 
894)66 
895984 

8-897596 

899203 

900803 

902398 

903987 

905570 

907147 

908719 
910285 
91 1846 

8-913401 

914951 

916495 

9180)4 

919568 

921096 

92261c 

924i36 

926640 

927150 

8.928658 
93oi 55 
931647 
933 1 34 
934616 
036093 
937606 
939032 
940494 

941962 

3o- 19 
30-07 
29.  o5 
29-82 
29-70 
29-58 
29-46 
29-35 
29-23 
29-11 
29-00 

28-88 

28-54 

28-43 

28-32 
28-21 
28- 1 1 
28-00 
27.89 

2:3 

27-58 

27-47 
27.37 
27-27 
27-17 
27-07 
26-07 
• 26-87 

26-77 
26-67 
26-58 
26-48 
26-38 
26-29 
26-20 
26- 10 
26-01 
26-92 

25- 83 

26- 74 
25-65 
25-56 
25-47 
25-38 
25-3o 
25-21 
25-12 

25 -o3 

24 -o5 
24-86 
24.78 
24-70 
24-61 
24-53 
24-45 
24-37 

24 -3o 
24-21 

1 1 • 1 55356 
153545 
151740 
149943 
148154 
146372 
144597 
142829 
141068 
139314 
137567 

11 -135827 
134094 
135368 
130649 

1 28936 
127230 

1 2553 1 
123838 

1 22 1 5 1 

1 20471 

11-118708 

1171)1 

1 1 6470 

1 1 38i 5 
112167 
iio524 
108888 
107258 
io5634 

1 0401 6 

11  102404 
100797 
099197 
097602 
096013 
094430 
092853 
091281 
089716 
088 1 54 

1 1 • 086699 
o85o4o 
o835o5 
081966 
080432 
078004 
077)81  | 
dp864 
07435i 

072844 

11  071342 
069845 
068353 
066866 
065384 
063907 
062435 
060968 
069006 
068048 

60 

£ 

|g 

1 55 

1 54 

53 

52 

5i 

5o 

2 

2 

45 

44 

43 

42 

4i 

4o 

& 

2 

35 

34 

33 

32 

3i 

3o 

3 

2 

25 

24 

23 

22 

21 

20 

2 

2 

i5 

U 

i3 

12 

11 

10 

8 

8 

5 

4 

3 

2 

0 

Coaine 

D.  1 Sino 

Cotang 

D. 

Tang. 

M. 

(85  DEGREES.) 


SINES  AND  TANGENTS.  (5  DEGREE., 


28 


M 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

b 940296 
9417J8 

24 -o3 

9.998344 

.19 

8-941952 

24-21 

r 1 • o58o48 

60 

i 

23-94 

998333 

19 

943404 

24- 13 

056696 

59 

a 

943i74 

23-87 

998322 

•19 

944852 

24 -o5 

o55i48 

58 

3 

944606 

23-79 

99831 1 

-19 

946205 

947784 

23-97 

1 o53"io5 

a 

4 

946034 

23-71 

998300 

•19 

28-90 

1 062266 

56  ! 

5 

947456 

940874 

950287 

23-63 

998289 

•19 

949168 

23-82 

o5o832 

55 

6 

23-55 

998277 

-19 

950597 

23-74 

o494o3 

54 

7 

23-48 

998266 

•19 

962021 

23-66 

047979 

53 

8 

951696 

23-4o 

998255 

-19 

953441 

23  • 60 

046559 

52 

9 

953 1 00 

23-32 

998243 

-19 

954856 

23-5i 

o45i44 

5i 

10 

954499 

23-25 

998232 

19 

956267 

23-44 

043733 

5o 

ii 

8- 955894 
957284 
958670 

23-17 

9 998220 

! -19 

8-967674 

23-37 

11  042326 

49 

ia 

i3 

23-10 

23-02 

998209 
998 1 97 

19 

•19 

969076 

960473 

23  -29 
23-23 

! 040925 

| 039027 

48 

47 

*4 

960002 

22-95 

22-88 

998 1 86 

-19 

96 1 866 

23-14 

o38i34 

46 

i5 

961429 

998174 

I -19 

963255 

23-07 

036745 

o3536i 

45 

16 

962801 

22- 80 

998163 

•19 

964639 

23-00 

44 

!2 

964170 

22-73 

22-66 

998i5i 

998139 

.19 
• 20 

966019 

o673o4 

| 22 -o3 

22-86 

o3398i 

o326o6 

43 

42 

7 I 7^ 

968766 

>9 

966893 

22-59 

998128 

•20 

1 33-79 

o3i234 

41 

ao 

968249 

22-52 

998116 

•20 

970133 

22-71 

029867 

40 

21 

8-969600 

22-44 

22-38 

9-998104 

•20 

. on 

8-971496 

972855 

22-65 

00  • 5-? 

1 1 -o285o4 
027 145 

998080 

m m si  j 

22  • 5l 

23 

972289 

22  -3l 

•20 

974209 

020791 

3] 

24 

973628 

22-24 

998068 

•20 

97556o 

22-44 

024440 

36 

25 

974962 

22-17 

998056 

•20 

976906 

22-37 

023094 

35 

26 

976293 

22-10 

998044 

-20 

978248 

22  -3o 

021702 

34 

37 

977619 

22  -o3 

998032 

•20 

979586 

22-23 

020414 

33 

28 

978941 

21*97 

998020 

•20 

980921 

22-17 

019079 

32 

29 

980269 

31  -9® 

998008 

•20 

982251 

22-10 

017749 

016423 

3i 

3o 

981673 

21-83 

997996 

•20 

983577 

22-04 

3o 

3! 

8-982883 

2i*77 

9.997985  ' 

•20 

8 • 984899 

21-97 

1 1 -oiSioi 

39 

32 

984189 

21 -7°  j 

99797 3 i 
997969  | 

•20 

986217 

21  -9I 

013783 

28 

33 

985491 

21-63 

•20 

987532 

988842 

21-84 

012468 

27 

34 

986789 

21-57  j 

997947  i 
997935 

•20 

21-78 

01 1 1 58 

26 

25 

35 

988o83 

21  -5o 

•21 

990149 

21  -71 

009851 

36 

989374  ! 

21-44 

997922 

•21 

991451 

21  -65 

008549 

24 

990660 

2i-38 

997010 

•21 

992750 

21-58 

007 2 5o 

23 

38 

991943 

21  -3i 

997807 

-21 

994045 

21-52 

000955 

22 

.39 

993222 

21-25 

997885 

•21 

995337 

21  -46 

004663 

21 

4o 

994497 

21  • 19 

997872 

•21 

996624 

21  -40 

003376 

20 

4i 

8-995768 

21-12 

9-997860 

•21 

8-997908 

21-34 

1 1 002092 

*9 

42 

997036 

998299 

21  06 

997847 

997830 

•21 

999188 

21-27 

000812 

10 

43 

21  -00 

•21 

9 • ooo465 

21-21 

10-999535 

998262 

17 

44 

999560 

20-94 

997822 

•2! 

001738  1 

21  • l5 

10 

45 

9000816 

20-87 

997809 

•21 

003007 

31  "°9 

996993 

i5 

46 

002060 

20-82 

997797 

-21 

004272 

oo5534 

21  -04 

995728 

U 

*7 

oo33io 

20-76 

997784 

-21 

20-97 

994466 

i3 

48 

004563 

20-70 

SBS 

•21 

006792 

20-91 

993208 

12 

49 

oo58o5 

20-64 

•21 

008047 

20-85 

991953 

II 

5o 

007044  | 

20-58 

997745 

• 21 

009298 

20- 80 

990702 

10 

5i 

9-008278 

20-52 

9.997732 

21 

9-010546 

20.74 

ic -989464 

9 

52 

009510  1 

20-46 

997719 

•21 

011700 

20-  08 

988210 

8 

53  I 

010737  | 

20  -4» 

997706 

997603 

•21 

oi3o3i 

20-62 

986969 

7 

54 

01196a  | 

20-34 

• 22 

014268 

20-56 

986732 

5 

55 

oi3i8a 

20-2Q 

997680 

997667 

• 22 

oi55o2 

20-  5l 

084408 

5 

56 

014400  1 

20-23 

•22 

016732 

20-45  j 

983268 

4 

h 

oi56i3 

20-17 

997654 

•22 

017950 

oi9i83 

20-40 

982041 

3 

58 

016824 

20-12 

997641 

•22 

20-33 

980817 

2 

59 

oi8o3i 

20  06 

997628 

•22 

020403 

20-28 

1 

60 

019235  | 

20-00 

997614 

•22 

021620 

20-23 

° 

r 

Cotine  1 

D.  1 

Sine 

Cotang. 

p. 

M.J 

(84  DEGREES.^ 


VG  DEGREES.)  A TABLE  OF  LOGARITHMIC 


44 


M 

Sine 

D. 

Cosine 

D. 

Tan*. 

1 D' 

Cotnnff. 

0 

9019235 

30-00 

9-997614 

-22 

9-021620 

20-23 

10-978380 

i 60 

i 

020435 

19-o5 

997601 

•22 

022834 

20-17 

977166 

975956 

5q 

2 

021632 

19-89 

997588 

•23 

024044 

30- II 

58 

3 

4 

022825 

024016 

19-84 

19-78 

997574 

997561 

•22 

•22 

025251 

026455 

20  • 06 
20-00 

$$ 

& 

5 

025203 

19-73 

997547 

•23 

, 027655 

1995 

972345 

55 

6 

026386 

19-67 

997534 

•23 

028853 

19-9° 

97**48 

54 

7 

027567 

028744 

19-62 

997520 

•23 

o3oo46 

19-85 

909954 

53 

8 

19-57 

997507 

997403 

997480 

•23 

o3i23t 

• 979 

968763 

5a 

9 

IC 

029918 

031089 

i95i 

19-47 

•23 

•23 

032425 

033609 

*9-74 

*9-69 

5i 

5o 

ii 

9-032257 

i9*4i 

9-997466 

-23 

9-034791 

19. 64 

10-966209 

4<? 

13 

o3342I 

19-36 

997452 

•23 

035969 

19-58 

96403 1 

48 

i3 

o34582 

I9-3o 

99743o 

997425 

-23 

037144 

I9-53 

- 962856 

47 

U 

035741 

036896 

19-25 

-23 

o383i6 

19.48 

961684 

46 

i5 

19-20 

997411 

-23 

039485 

19-43 

9605 1 5 

45 

16 

o38o48 

I9-i5 

997307 

997383 

-23 

o4o65 1 

i9-38 

969349 

44 

•7 

039197 

19-10 

-23 

041813 

19-33 

968187 

43 

18 

•9 

040342 

041485 

io-°5 

18-99 

•23 

•23 

042973 
044 i3o 

19-28 

19-23 

967027 

966870 

42 

41 

20 

042625 

18-94 

997341 

•23 

046284 

19- 18 

964716 

40 

31 

9-043762 

18-89 

999732] 
9973 1 3 

• 24 

9 • 046434 

19- 13 

10-953566 

23 

044895 

18-84 

•24 

047582 

19-08 

952418 

38 

23 

046026 

18-79 

697209 

•24 

048727 

19-03 

95i2]3 

37 

24 

047154 

18-75 

997285 

•24 

049860 

18-98 

95oi3i 

36 

1 25 
26 

048279 

049400 

18-70 

1 8 - o5 

997271 

997267 

•24 

•24 

o5iooo 

o52i44 

i8-o3 

18-89 

948992 

947856 

35 

34 

\l 

o5o5i9 

o5i635 

18-60 

18-55 

997242 

997228 

•24 

•24 

053277 

054407 

18-84 

18-79 

946723 

945593 

33 

32 

29 

062749 

o53859 

i8-5o 

997214 

•24 

055535 

18-74 

944465 

3i 

3o 

18-45 

997199 

•24 

056659 

18-70 

943341 

3o 

3i 

9 • 054966 

18-41 

9-997*85 

•24 

9-057781 

18. 65 

10-942219 

29 

32 

05607 1 

i8-36 

997170 
997 1 56 

•24 

058900 

*8-69 

941100 

28 

33 

057172 

1 8 - 3 1 

•24 

060016 

18-55 

939984 

938870 

27 

34 

068271 

18-27 

997*41 

•24 

061 i3o 

18. 5i 

26 

35 

059367 

18-  22 

997*27 

•24 

062240 

18-46 

25 

36 

060460 

i8-17 

997**2 

•24 

063348 

18-42 

24 

37 

o6i55i 

18- i3 

997008 

997083 

•24 

064453 

18-37 

935547 

23 

38 

062639 

18-08 

-25 

o65556 

i8-33 

934444 

22 

39 

063724 

064806 

18-04 

997068 

•25 

o66655 

18-28 

933345 

21 

4o 

17-99 

997053 

•25 

067752 

18-24 

932248 

20 

4i 

9-o65885 

17-94 

9.997039 

•25 

9 • 068846 

18-19 

io-93ii54 

*9 

42 

066962 

17 -oo 

997024 

-25 

069938 

i8-i5 

930062 

18 

43 

o68o36 

17-86 

997009 

•35 

071027 

18- 10 

928973 

*7 

44 

069107 

17-81 

996994 

•25 

072113 

18 -06 

16 

i5 

45 

070176 

17.77 

996979 

•35 

073197 

074278 

18-02 

46 

071242 

17-72 

996964 

1 - 25  I 

*7*97 

925722 

14 

47 

072306 

17-68 

996949 

■35  ! 

075356 

!7*o3 

924644 

i3 

48 

073366 

17*63 

296934 

-25 

076432 

17.89 

923568 

12 

49 

074424 

17 -5o 

996919 

•25 

o7]5o5 

078676 

17-84 

922495 

11 

5o 

075480 

^•55 

996904 

•25 

17-80 

921424 

10 

5i 

9-076533 

17  -5o 

9-996889 

•25 

9-079644 

080710 

081773 

17.76 

10 -920356 

9 

53 

53 

077583 

07863 1 

17-46 

17-42 

996874 

996868 

•25 

•25 

*7-72 

17-67 

919290 

918227 

8 

7 

54 

55 

079676 

080719 

17-38 

1 7 • 33 

996843 

996828 

•25 

•25 

082833 

083891 

17-63 

*7-59 

917*67 

9*6*00 

0 

5 

081759 

17-29 

9968 1 2 

• 26 

084947 

17  55 

9i5o5j 

4 

3 

082707 

083832 

17-25 

17-21 

996707 

996782 

• 26 

• 26 

086000 

087050 

17  5i 
*7  47 

9 1 4000 
912950 

3 

2 

59 

084864 

*7-«7 

996766 

.26 

088098 

I7-4J 

91 1002 
910856 

1 

1 60 

08 5894 

17  ii 

996761 

• 26 

089144 

•7-38  J 

0 

1“ 

< faine 

D. 

Sinn 

Cotfing. 

D.  1 

mT 

(83  DEGRKR8.) 


SINES  AND  TANGENTS.  (7  DEGREES.) 


26 


ML 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

60 

Is 

3 

55 

54 

53 

52 

5i 

5o 

S 

2 

45 

44 

43 

42 

4i 

4o 

Is 

35 

34 

33 

32 

3i 

3o 

22 

22 

25 

24 

23 

22 

21 

20 

22 

22 

i5 

14 

i3 

12 

11 

10 

2 

2 

5 

4 

3 

2 

0 

0 

1 

a 

3 

4 

5 

6 

2 

9 

10 

11 

12 
i3 
U 

15 

16 

22 

*9 

20 

21 

22 

23 

34 

25 

26 

3 

IS 

31 

32 

33 

34 

35 

36 

ll 

39 

40 

41 
43 

43 

44 

45 

46 

% 

ft 

s' ' 

52 

53 

54 

55 

56  | 

U 

Is 

9 • 085894 
086922 
081947 
088970 
089990 
091008 
092024 
093037 
09404] 
095o56 
096062 

9-097065 

098066 

099065 

100062 

ioio56 

102048 

io3o37 

io4o25 

io5oio 

106992 

9-106973 

107951 

108927 

109001 

110873 

111842 

112809 

113774 

114737 

1 1 5698 

9-ii6656 

117613 

118567 

119519 

1 20469 
121417 
122362 

1 233o6 
124248 
125187 

9-126126 

127060 

127993 

128925 

1 29854 
130781 
i3ijo6 
i3263o 
i3355i 
134470 

9-13538] 

i363oj 

137216 

1 38128 
139037 
139944 

i4o85o 

i4n54 

142655 

143555 

17  • i3 
17.09 

17-04 

17-00 

16-96 

16-92 

16.88 

16-84 

16-80 

16-76 

16-73 

16.68 

i6-65 

16  • 61 

16 -5] 
i6-53 
16-49 

i6-45 

16-41 

i6-38 

i6-34 

i6-3o 

16-27 

i6-23 

16-19 

16-16 

16-12 

16-08 

i6-o5 

1601 

1 5 • 97 

i5-94 

15-90 

1 5 - 87 
i5-83 

15- 8o 
i5  - 76 

1 5 - 73 

16- 69 

1 5- 66 
i5-62 

i5-59 

1 5 • 56 

1 5 • 62 
i5-49 
15-45 
i5-42 

1 5 • 39 

15- 35 
i5  32 

16- 29 

1 5 - 25 

15-  22 

16- 19 
i5  - 16 

1 5 - 1 2 
i5-09 
i5-o6 

15- o3 

16- 00 
14-96 

9.996751 

996735 

996720 

996704 

996688 

996673 

996667 

996641 

996625 

996610 

996594 

9-996578 

996562 

996646 

996530 

996614 

996498 

996482 

996465 

996449 

996433 

9-996417 

996400 

996384 

996368 

99635i 

996335 

996318 

996302 

996285 

996269 

9-996252 

996235 

996219 

996202 

996185 

996168 

996151 

996134 

996117 

996100 

9-996083 

996066 

996049 

996032 

996015 

9959o8 

995980 

995963 

995946 

996928 

9*9959” 

996894 

995876 

996859 

995841 

995823 

995806 

995788 

$$ 

• 26 
•26 

• 26 

• 26 

• 26 
-26 

• 26 

• 26 

• 26 
•26 

• 26 

.27 

•27 

.27 

•27 

•27 

•27 

•27 

.27 

•27 

•27 

•27 

•27 

•27 

•27 

•27 

•27 

:22 

.28 

•28 

.28 

.28 

.28 

.28 

.28 

.28 

• 28 
-28 
.28 
.28 

•29 

•29 

•29 

•29 

•29 

•29 

•29 

•29 

•29 

•29 

29 

•29 

•29 

• 29 

•29 

•29 

• 29 

• 29 
•29 
•29 

9 089:44 

090187 
091 225 
092266 
093302 
094336 
095367 
096395 
097422 
098446 
099468 

9- 100487 
101604 
102519 
io3532 
104542 
io555o 
io6556 
107559 
io856o 
109559 

9- iio556 

111 55 1 

1 1 2643 

1 1 3533 
114521 

1 1 55o7 

1 16491 
H7472 
118452 
119429 

9-120404 

Will 

123317 

124284 

125249 

126211 

1 27172 
i28i3o 

1 29087 

9- i3oo4i 

1 3 0994 

131944 

132893 

133839 

134784 

135726 

1 3666] 
137605 

1 38542 

9-139476 

140409 

i4i34o 

142260 

143190 

144121 

1 46044 
145966 
146885 
147803  | 

17-38 

17.34 

i7-3o 

17.27 

17-22 

1710 

17  • i5 

17*11 

17-07 

17-03 

16-99 

16-96 

16-91 

16-87 

16-84 

16-80 

16-76 

16-72 

16-09 

i6-65 

16-61 

i6-58 
i6-54 
i6-5o 
16-46 
i6-43 
16-39 
i6- 36 
i6-32 
16-29 
i6-25 

16  • 22 
16. 18 
i6-i5 
16- 1 1 
16-07 
16-04 
16-01 
iJ-97 

15- 94 

16- 91 

15.87 

16-84 

1 5 • 81 

1 5 - 77 

1 5 • 74 

16-71 

16-67 

16-64 

1 5 -61 

1 5 • 58 

1 5- 55 
i5-5i 
i5-48 
i 5 - 45 

1 5 • 42 

1 5 - 39 

1 5 - 3d 

1 5 • 32 
i5-29 
i5-26 

10-9108 56 
909813 
908772 

907734 

90669b 

905664 

904633 

9o36o5 

902578 

901554 

9oo532 

10-899513 

898496 

897481 

896468 

896458 

894450 

893444 

892441 

891440 

890441 

10-889444 

888449 

887457 

886467 

885479 

8844Q^ 

8835o9 

882528 

881548 

880671 

10-879696 

878623 

877652 

876683 

875716 

874751 

sga 

871870 

870913 

10-869959 

869006 

868o56 

867107 

866161 

865216 

864274 

863333 

862395 

861458 

io-86o524 

859591 

85866o 

857731 

85o8o4 

855879 

854950 

854o34 

853ii5 

852197 

1 

Coeine 

D.  1 Sine 

Ootftng.  D. 

_ Tang. 

M. 

(82  DEGREES.) 


v8  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 

Sine 

D- 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9> 143555 

! 14-96 

9-995753 

•3o 

9 • 1 47803 

i5-a6 

10852197 

60 

i 

144453 

14-93 

995735 

•3o 

148718 

1 5- 23 

85ia8a 

j 69 

a 

145349 

14-90 

995717 

•3o 

149832 

|5*20 

85o368 

! 58 

3 

146243 

i47i36 

1 14-87 

- 3o 

1 5o544 

1 5i 454 

1 *.n 

849456 

848546 

IS 

4 

14-84 

yyjuuy 

990681 

•3o 

*7 

1 5 • 1 4 

5 

148026 

1 4 • 81 

995664 

•3o 

152363 

1 5 • 1 1 

847637 

846731 

845626 

1 55 

6 

148915 

14-78 

995646 

•3o 

163269 

i5-o8 

54 

7 

149002 

14-75 

995628 

•3o 

i54i74 

i5-o5 

53 

6 

1 5o686 

14-72 

995610 

•3o 

1 55077 

i5oa 

844923 

1 5a 

9 

161569 

14-69 

995591 

•3o 

155976 

166877 

14-99 

844022 

i 5i 

IO 

1 5245 1 

14-66 

995573 

•3o 

14-0 

843123 

! 5o 

i. 

9-1  333o 

14-63 

9-995555 

•3o 

9-167775 

158671 

14-93 

10-842226 

49 

la 

1 54208 

14-60 

995537 

•3o 

14-90 

841329 

48 

i3 

i55o83 

14*57 

995519 

-3o 

1 69565 

14-87 

1 840436 

■ 4] 

14 

15 

155957 

1 5683o 

14-54 

1 4 • 5i 

995501 

995482 

•3i 

•3i 

160457 

161347 

14-84 

14*81 

839543 

838653 

i 46 
45 

16 

167700 

14-48 

995464 

•3i 

162236 

*4-79 

837764 

836677 

44 

*7 

1 58069 

14-45 

995446 

•3i 

i63i 23 

14-76 

43 

id 

159435 

14-42 

995427 

•3i 

164008 

14*73 

835992 

42 

>9 

i6o3oi 

14-39 

995409 

•3i 

164892 

14-70 

835 1 08 

4i 

ao 

161164 

14-36 

995390 

•3i 

165774 

14*67 

834226 

40 

ai 

9- 162025 

14-33 

9-995372 

995353 

• 3i 

9- 1 66654 

14-64 

io-833346 

39 

aa 

162885 

i4-3o 

• 3i 

167532 

14  • 61 

832468 

38 

a3 

163743 

164600 

14-27 

995334 

• 3i 

168409 

14-58 

831691 

j 37 

U 

1424 

9953i6 

•3i 

169284 

14-55 

830716 

1 36 

a5 

166454 

14-22 

& 

996260 

• 3i 

170157 

14-53 

829843 

35 

a6 

166307 

14-19 

•3i 

171029 

14 -5o 

828971 

| 34 

27 

167 i5o 
168008 
168856 

14*16 

•3i 

171899 

14-47 

828101 

33 

28 

39 

I4*i3 

i4-io 

995241 

995222 

-32 

-32 

172767 

173634 

14-44 

i4-4a 

827233 

828366 

3a 

3i 

3o 

169702 

14-07 

995203 

•32 

174499 

14-39 

8255oi 

3o 

3i 

9-170547 

i4-o5 

9-995l84 

•32 

9- 175362 

14*36 

10-824638 

29 

3a 

171389 

14-02 

995 i 65 

-32 

176224 

14-33 

823776 

28 

33 

172230 

i3  -99 

996146 

-32 

177084 

i4-3i 

822916 

37 

34 

35 

173070 

173908 

13-96 

i3-94 

995127 

995108 

•32 

•32 

177942 

178799 

1 79655 

1 6o5o8 

14-28 

i4-25 

822068 

821201 

28 

25 

36 

174744 

175578 

i3-oi 

996089 

-32 

14-23 

820345 

24 

37 

13-88 

995070 

-32 

14-20 

819492 

818640 

23 

38 

17641 1 

i3  • 86 

995061 

-32 

i8i36o 

i4-i] 

22 

3g 

177242 

1 3-83 

995o32 

•32 

182211 

i4- 15 

817789 

21 

40 

178072 

i3-8o 

995oi 3 

•32 

i83o59 

1412 

816941 

20 

41 

9- 178900 

1 3 - 77 

9.994993 

-32 

9- 183907 

14-09 

10-816093 

42 

43 

Xlt 

1 3 - 74 
i3 • 72  n 

994974 
».  994955 

•32 

•32 

184752 

i855o7 

1 86489 

14-07 

14-04 

8i5248 

8i44o3 

18 

44 

181 37.' 

13-69 

994935 

•32 

14-02 

8i356i 

16 

45 

182196 

1 3 • 66 

994016 

994896 

.33 

187280 

1 3 *99 

812720  1 

i5 

46 

i83oi6 

1 3 -64 

.33 

188120 

13*96 

81 1880  ! 

14 

47 

183834 

i3-6i 

994877 

994857 

.33 

i88958 

i3 -93 

811042  I 

i3 

40 

1 8465i 

i3-5^  / 

•33  1 

189794 

13-91 

810206  1 

12 

49 

185466 

i3-5b 

994838 

• 33  | 

190629 

13-89 

80937 1 ! 
8o8538 

11 

5o 

186280 

i3-53 

994818 

• 33 

191462 

i3-86 

10 

5i 

9' 18-092 

i3-5i 

9.994798 

•33 

9-192294 

i3-84 

10-807706 

9 

5a 

187903 

188712 

1 3 - 48 

994779 

•33 

193124 

;3-8i 

806876 

8 

53 

1 3 • 46 

994709 

• 33 

193953 

1 3 - 70 

806047 

7 

54 

55 

i8q5io 

190J2D 

1 3 • 43 

1 3 - 4 1 

994739 

994719 

• 33 
•33 

194780 

196606 

i3  • 76 
i3  • 74 

805220 

804394 

6 

5 

56 

191130 

i3-38 

994700 

994680 

994660 

•33 

196430 

i3-ji 

803570 

4 

3 

191933 
192734 
193534  1 

1 3 • 36 

1 3 • 33 

•33 

•33 

197253 

198074 

1369 

1 3 • 66 

802747 

801926 

3 

i 

h 

i3-3o 

994640 

• 33 

198894 

1 3 • 64 

801 106 

1 

60 

■9433a  | 

i3-28 

994620 

• 33 

199713 

1 3 • 61 

800287 

0 

Comne  | 

D.  1 

Sino 

Cotang. 

D. 

Tang. 

JL 

(81  DEORKEO.) 


SINES  AND  TANGENTS.  (9  DEGREE.) 


27 


< M. 

Bine 

D. 

Cosine 

D. 

Tang. 

D. 

| Cotang. 

r 1 

0 

q- 194332 

i3-a8 

9-994620 

•33 

9-199713 

i3-6i 

10-800287 

1 to 

a 

i95i 20 
l9592D 

1 3 • 26 

1 3 * 23 

994600 

99458o 

•33 

•33 

2oo5ao 

2oi345 

1 3 • 59 

1 1 3 • 56 

799471 

798655 

s 

3 

196719 

1 3 - 2 1 

99456o 

34 

202 i59 

: 3 • 54 

797841 

, 67 

4 

19731 I 

1 3 - 1 8 

994540 

•34 

202971 

1 3 • 52 

797029 

! 56 

5 

i983o2 

1 3 • 1 6 

994519 

• 34 

203782 

1 1 3 • 49 

7962 1 8 

55 

6 

1 9909 1 

1 3 - 1 3 

99449^ 

•34 

204592 

| i3  * 47 

795408 

1 54 

7 

199879 

1 3- 1 1 

994479 

•34 

205400 

i3-45 

7Q4600 

! 53 

6 

200666 

i3-o8 

99445o 

•34 

206207 

i3-42 

793703 

5a 

9 

201431 

i3-o6 

994438 

•34 

20701 3 

i3-4o 

79298] 

| 792183 

1 5i 

10 

202234 

i3-o4 

9944i8 

•34 

207817 

1 3 - 38 

; 50 

11 

9-203017 

i3-oi 

9-994397 

•34 

j 9-208619 

1 i3 - 35 

io-79i38i 

: 49 

is 

i3 

303797 

204377 

1299 

12-96 

994377 

994357 

•34 

•34 

209420 

210220 

1 1 3 • 33 

! i3 - 3 1 

790580 

589780 

! 48 

47 

• 4 

205354 

12-94 

994336 

•34  1 

2lIOl8 

i i3-28 

788982 

46 

i5 

2o6i3i 

12-92 

9943 1 6 

•34 

2 I 1 81 5 

1 i3-26 

788186 

45 

l6 

206906 

12-89 

994295 

•34  j 

212611 

1 3 - 24 

787389 

786595 

785802 

44 

\l 

201679 

208402 

12-87 

12-85 

994274 

994254 

•35 

•35 

2i34o5 

214198 

1 3 - 2 1 
i3  - 19 

43 

42 

209222 

12-82 

994233 

•35 

2 14989 

i3- 17 

785oi 1 

41 

30 

209992 

12 -8o 

994212 

•35 

215780 

i3- 15 

784220 

40 

31 

9 210760 

12-78 

9-994191 

•35 

9- 2i6568 

i3- 12 

10-783432 

3q 

3a 

211526 

12-75 

994171 

994 1 5o 

•35 

217356 

i3- 10 

782644 

38 

33 

212291 

12-73 

-35 

218142 

i3-o8 

781858 

3] 

34 

35 

2i3o55 

2 i38i8 

12-71 

12-68 

994120 

994108 

-35 

-35 

218926 

219710 

i3-o5 

i3-o3 

781074 

780290 

36 

35 

36 

27 

214579 

2i5338 

12-66 

12-64 

994o8i 

994066 

• 35 
-35 

220492 

221272 

i3-oi 

12-99 

7795o8 
778728  | 

34 

33 

38 

216097 

12-61 

994045 

•35 

222052 

12-97 

777948  i 

32 

29 

216854 

I2-59 

994024 

•35 

222830 

12-94 

777170  ; 

3i 

3o 

217609 

12-67 

994oo3 

•35 

2236o6 

12-92 

776394  : 

3o 

3i 

9- 2i8363 

12-55 

9-993981 

-35 

9-224382 

1 2 -90 

10-775618 

29 

3a 

219116 

12-53 

99396o 

• 35 

225i56 

12-88  I 

774844 

28 

33 

219868 

I2-5o 

993939 

993918 

993896 

-35 

225929 

12-86  1 

774071 

27 

34 

220618 

12-48 

-35 

226700 

12-84  | 

773300 

20 

35 

221367 

12-46 

-36 

227471 

228239 

12-81  j 

772529 

25 

36 

2221 13 

13-44 

993876 

993854 

-36 

12-79 

771761 

34 

h 

222861 

12-42 

-36 

229007 

12-77 

770993 

23 

38 

223606 

I2-39 

993 83 2 

-36 

229773 

12-75 

770227 

22 

39 

224349 

12-37 

99381 1 

• 36 

23o539 

12-73 

769461 

21 

4o 

225o92 

12*35 

993789 

• 36 

23i3o2 

12-71 

768698 

20 

4i 

9-225833 

12-33 

9-993768 

•36 

9- 232o65 

12-69 

10-767935 

*9 

43 

226673  ! 

1 2 - 3 1 

993746 

•36 

232826 

12-67  I 

767174 

766414 

l8 

43 

227311 

12-28 

993726 

-36  ' 

233586 

ia-65  1 

17 

44 

228048  1 

12-26 

993703 

•36 

234345 

12-62  1 

765655 

IO 

45 

328784  ! 

12-24 

99368i 

-36 

235 1 o3 

12-60 

764897 

i5 

46 

2295i8  , 

12-22 

99366o 

• 36 

235S39 

12-58 

764141 

14 

*7 

230252 

12-20 

993638 

-36 

2366i4 

12-56 

763386 

1 3 

48 

23o984 

12-18 

9936i6 

•36 

23i368 

12-54 

762632 

1; 

49 

231714 

I2-l6 

993594 

• 37 

238120 

12-52 

761880 

11 

5o 

232444 

12-14 

i 993572 

• 37 

238872 

12 -5o 

761128 

1 f° 

5. 

7 233 172 

12  12 

9-99355o 

.37 

9-239622 

12-48 

10-760378 

9 

5s 

233899 

12-09 

993528 

•37 

240371 

12-46 

769629 

! 8 

53 

334633 

13-07 

993 5o6 

•37 

241118 

12-44 

758882 

1 7 

54 

235340 

13-03 

993484 

•37 

241865 

12-42 

758i35 

6 

55 

23607J 

1 2 • o3 

993462 

• 37 

242610 

12-40 

757390 

5 

56 

236795 

i 12-01 

I 093440 

• 37 

243354 

ia-38 

756646 

4 

237015 

I *1-99 
11-97 

1 9934 1 8 

•37 

244097 

ia-36 

755qo3 

3 

58 

338235 

1 993396 

1 37 

2448)9 

12-34 

755i6i 

, 3 

1 *9 

a38953 

1 1 95 

! 993374 

1 993351 

•37 

245579 

12-32 

764431 

1 « 

|_6o_ 

339670 

11-93 

,37 

2463 1 9 

I2-3o 

7 5368 1 

1 0 

:! 

Coaino 

L. 

Sine  | 

Cotang. 

i_  JD. 

! Tang. 

M. 

(80  DEGREES.) 


(10  DEGREES.)  A TABLE  OF  LOGARITHMIC 


28 


M. 

Sino 

D. 

Cowine 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9 239670 
240386 

1 1 -93 

9993351 

•37 

9-246319 

12 -3o 

io-75368i 

60 

i 

11-91 

993329 

.37 

247067 

12-28 

752943 

5o 

2 

3 

241 101 

241814 

11-89 

11-87 

n-85 

993307 

993285 

I7 

■ 37 

28S 

12-26 

12  24 

752206 

751470 

1 750786 

58 

57 

4 

242526 

993262 

.37 

249264 

12-22 

56 

5 

243237 

n-83 

993240 

•37 

249908 

250780 

251461 

12-20 

1 750002 

55 

6 

7 

243o47 

244o56 

11-81 

11-79 

■*•77 

993217 

993196 

-38 

-38 

12-18 

1 2 ' * 7 

I 2 • 1 5 

1 ggg 

54 

53 

8 

245363 

993172 

-38 

262191 

747809 

52 

9 

246069 

246775 

11-75 

11-73 

993i49 

-38 

252920 

12  - 13 

747080 

5i 

10 

993127 

-38 

t 253648 

12-11 

746352 

5o 

li 

9-247478 

n-71 

9-993104 

•38 

9-254374 

12-09 

10-745626 

3 

la 

248181 

11-69 

99308 1 

•38 

255ioo 

1 -12-07 

12-05 

12-03 

744900 

13 

14 

248883 

249583 

1 1 -67 

1 1 -65 

993o5o 

993o3o 

-38 

•38 

255824 

256547 

744n6 

743453 

3 

i5 

250282 

n-63 

99301 3 

-38 

257269 

12-01 

742731 

45 

16 

260980 

1 1 -6i 

992990 

-38 

12-00 

742010 

44 

*7 

2523j3 

253067 

11-59 

11-58 

n-56 

992967 

•38 

1 I -98 

741290 

43 

18 

'9 

992944 

•38 

-38 

259429 

260146 

1 1 -96 
11*94 

740571 

739854 

42 

41 

20 

253761 

n-54 

■ 38 

260863 

1 1 -92 

739*37 

40 

21 

9-254453 

1 1-52 

9-992876 

992852 

• 38 

9-261678 

1 1 -90 

10-738422 

3g 

22 

255i44 

11  -5o 

•38 

262292 

11-89 

737708 

38 

23 

255834 

11-48 

992820 

992800 

-39 

263oo5 

11-87 

736995 

37 

U 

256523 

11-46 

-39 

263717 

264428 

11-85 

736283 

36 

^25 

257211 

ii-44 

992783 

• 39 

n-83 

735572 

35 

26 

2 

257898 

258583 

11-42 

11  -4i 

992759 

992736 

I9 

• 39 

265 1 38 
265847 
266555 

11-81 

11-79 

734862 

734i53 

34 

33 

259268 

11-39 

992713 

•39 

11-78 

733445 

3a 

£ 

269951 

a6o633 

11-37 

11-35 

992600 

992666 

-39 

-39 

267261 

267967 

11-76 

ii-74 

3i 

3o 

3i 

9-a6i3i4 

n-33 

9-992643 

-39 

9-268671 

11-72 

10-731329 

730625 

3 

32 

261994 

262673 

263351 

ii-3i 

992619 

-39 

269375 

11-70 

33 

11  -3o 

992596 

-39 

270077 

11-69 

729923 

3 

34 

11-28 

992572 

.39 

270779 

1 1 -67 
n-65 

729221 

728521 

35 

264027 

11-26 

992049 

992525 

•39 

271479 

272178 

25 

36 

264708 

266877 

11-24 

-39 

11-64 

727822 

24 

37 

11-22 

992501 

•39 

272876 

11-62 

727124 

23 

38 

266061 

11-20 

992478 

.40 

273543 

1 1 -6o 

726427 

22 

39 

266723 

267895 

II-I9 

992454 

•40 

274269 

n-58 

725731 

21 

4o 

H*I7 

992430 

• 40 

274964 

11-57 

725o36 

20 

4i 

9 • a68o65 

1 1 - 15 

9-992406 

•40 

9-275658 

u-55 

10-724342 

:? 

4a 

268734 

1 1 - 13 

992382 

•40 

276351 

n-53 

723649 

43 

269402 

1 1 • 1 1 

992350 
992335 
9923ii  1 

• 40 

277043 

1 1 - 5i 

722961 

722266 

721676 

720887 

*7 

44 

45 

270069 

270736 

1 1 - 10 

11  -08 

•40 

I *4o 

277734 

278424 

1 1 - 5o 

11-48 

16 

i5 

46 

271400 

11  -06 

992287  1 
992263 

•40 

279113 

n-47 

14 

8 

49 

272064 

1 1 -o5 

.40 

279801 
280488 
281 174 

1 1 -45 

1 1 -43 
ii-4i 

720199 

i3 

272726 

273388 

1 1 -o3 

1 1 -01 

992239 

992214 

•40 

•40 

719512 

718826 

12 

1 1 

5o 

274049 

10-99 

992190 

•40 

281808^ 

n-40 

718142 

10 

5i 

9-274708 

275867 

10-98 

9-992166 

•40 

9-282542 

II-38 

10-717458 

716775 

8 

5a 

10-96 

992142 

•40 

283225 

n-36 

53 

276024 

10-94 

9921 17 

•4i 

283907 

284588 

u-35 

716093 

7 

54 

276681 

10-92 

992093 

992069 

•41 

n-33 

716412  1 

6 

55 

277337 

10-91 

10-89 

10-87 

10-86 

10-84 

10-82 

•41 

285268 

1 1 -3i 

714732 

5 

56 

57 

*7799' 

278644 

992044 

992020 

•4i 

•4i 

285947 

286624 

1 1 - 3o 
11-28 

7i4o53 

713376 

4 

3 

58 

59 

60 

279297 

279048 

280699 

991996 

99*97* 

991947 

•4i 

•4i 

•41 

287301 

11-26 

11-25 

11-23 

712690 

712023 

7 1 1 348 

2 

0 

Ooh’iiio 

I). 

Bine 

Cotang. 

I). 

Tttnfr_ 

M. 

(79  DKGRKK8.) 


SINKS  AND  TANGENTS.  (11  DEGREES.) 


29 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9-280599 

281248 

10-82 

9 99*947 

•4i 

9-288652 

11-23 

10-711348 

60 

i 

10-81 

991922 

•4i 

289326 

11-22 

710674 

5q 

j 

281897 

10-79 

991897 

•4i 

289999 

11-20 

710001 

58 

3 

282544 

10-77 

991873 

•4i 

29067 1 

1 1 • l8 

709329 

57 

4 

283190 

10-76 

991848 

•4i 

291342 

11*17 

708658 

56 

5 

283836 

io-74 

991823 

•4i 

292013 

II-I& 

707087 

707818 

55 

6 

284480 

10-72 

991799 

•4i 

292682 

ii  -14 

54 

7 

285124 

10-71 

99*774 

.42 

29335o 

II  -12 

7o665o 

53 

6 

285766 

10-69 

991749 

•42 

! 294017 

II-II 

7o5o83 

705816 

52 

9 

286408 

10-67 

991724 

•42 

294684 

1 1 09 

5i 

10 

11 

12 

287048 

10-66 

10-64 

io-63 

991699 

9-99*674 

991649 

•42 

295349 

9- 296013 
296677 

11-07 

704661 

5o 

288326 

•42 

•42 

1 1 -06 

11  -04 

1 0 • 703981 

703828 

3 

i3 

288964 

io-6i 

991624 

•42 

297339 

298001 

1 1 o3 

702661 

47 

14 

289600 

io-5o 

991599 

•42 

1 1 -01 

7°* 99Q 
701888 

46 

i5 

290236 

io-58 

991574 

•42 

298662 

I I - 00 

45 

16 

290870 

io-56 

99*649 

•42 

299322 

10-98 

700678 

44 

*7 

291604 

io-54 

991624 

•42 

299980 

10-96 

700020 

43 

18 

292137 

292768 

293899 

io-53 

99*498 

•42 

3oo638 

10-95 

699362 

42 

•9 

io-5i 

99*473 

•42 

3oi2o5 

301961 

10-93 

698705 

4i 

20 

io-5o 

99*448 

•42 

10-92 

698049 

4o 

21 

9-294029 

10-48 

9 99*422 

•42 

9-302607 

10-00 

10-697393 

696789 

696086 

3q 

22 

294658 

10-46 

99*397 

•42 

3o326i 

10-89 

38 

23 

295286 

io-45 

991372 

•43 

3o3qi4 

304667 

10-87 

37 

24 

296013 

io-43 

991346 

•43 

io-86 

695433 

36 

25 

296639 

10-42 

991321 

•43 

306218 

10-84 

694782 

35 

26 

297164 

10-40 

991296 

•43 

3o5869 

io-83 

694131 

34 

J7 

297788 

10-39 

991270 

-43 

3o65i9 

io-8i 

693481 

33 

28 

• 298412 

10-37 

99*244 

•43 

307168 

io-8o 

692832 

32 

*9 

299034 

io-36 

991218 

•43 

307815 

3o8463 

10-78 

692185 

3i 

3o 

299665 

io-34 

991193 

•43 

10-77 

691537 

3o 

3i 

9-300276 

10-32 

9-991167 

•43 

9*309109 

10-75 

10-690891 

29 

32 

300895 

io-3i 

99**4i 

•43 

309764 

310898 

10.74 

690246 

28 

33 

3oi5i4 

10-20 

991116 

•43 

10-73 

689602 

27 

34 

302132 

10-28 

991090 

991064 

•43 

3iio42 

10-71 

688958 

26 

35 

302748 

3o3j64 

10-26 

• 43 

3 1 i685 

10-70 

6883 1 5 

25 

36 

10-25 

99io38 

•43 

3 1 2327 

io-68 

687673 

687083 

24 

37 

3o3o7o 

10-23 

991012 

•43 

312967 

3 1 3608 

10-67 

23 

38 

304398 

10-22 

990986 

•43 

io-65 

686392 

685753 

22 

39 

3o5207 

10-20 

990960 

•43 

314247 

314886 

10-64 

21 

4o 

3o58i9 

10-  19 

990934 

•44 

10-62 

685i 1 5 

20 

41 

9-3o643o 

10-  17 

9-990908 

990882 

•44 

9*3i5523 

io-6i 

10-684477 

‘9 

42 

307041 

I0-I6 

•44 

3i6i5o 

io-6o 

68384i 

18 

43 

307650 

10-  14 

990855 

•44 

316796 

io-58 

683205 

*7 

44 

308259 

io- 13 

990829 

•44 

317480 

10-57 

682570 

16 

45 

308867 

IO-II 

990808 

•44 

3 1 8064 

io-55 

681986 

i5 

46 

47 

309474 

3 1 0080 

10-10 

10-08 

990777 

990750 

•44 

•44 

318697 

319829 

io-54 

io-53 

681803 

68067 1 

14 

i3 

48 

3io685 

10-07 

990724 

990697 

•44 

319961 

io-5i 

680089 

12 

49 

3ii28o 

10-03 

•44 

320692 

io-5o 

679408 

11 

5o 

3 1 1 898 

10-04 

990671 

•44 

321222 

10-48 

678778 

10 

5i 

9 - 3 1 2495 

io-o3 

9-990644 

•44 

9*32i85i 

io-47 

10-678149 

9 

52 

53 

3 1 3097 
313698 

10-01 

10-00 

990618 

990591 

99o565 

•44 

•44 

322479 

323 106 

io-45 

io-44 

677521 

676894 

8 

7 

54 

314297 

9-98 

•44 

323733 

io-43 

676267 

0 

55 

314897 

3 1 5493 

997 

990538 

•44 

324858 

10-41 

676642 

5 

56 

9.96 

9905 1 1 

•45 

324983 

10-40 

676017 

4 

*7 

3 1 6002 
316689 

9.94 

99048 ) 

•45 

326607 

10-39 

674308 

3 

58 

9-93 

990458 

•45 

326231 

10-37 

673769 

2 

J9 

317284 

9-91 

990431 

•45 

326853 

io-36 

673147 

672625 

1 

60 

317879 

9.90 

99°4 J4 

•45 

327475 

io-35 

0 

I Cosine 

D. 

1 Sine 

Cotang. 

D. 

Tang, 

M. 

(78  DEOKEEB.) 


60 


(12  DEGREES., 


A TABLE  OF  LOGARITHMIC 


M. 

8ine 

D. 

Cosine 

1 

Tan*. 

D. 

Cotang. 

0 

9*317879 

3 1 8473 
319066 

99° 

9*990404 

1 *45 

9*327474 

328095 

1 328715 

1 io*35 

10*672526 

1 60 

i 

J 

9*88 

9*87 

990378 

99o35i 

I ^ 

1 -45 

10*33 

10*32 

671905 

671285 

3 

319668 

9*86 

990324 

*45 

329334 

io*3o 

1 670666 

1 57 

: i 

320249 

9*84 

990297 

! • 45 

329953 

33007c 

10*20 

1 670047 

56 

1 5 

320840 

0*83 

990270 

.45 

10*28 

1 669430 

55 

6 

32i43o 

9*82 

990243 

•45 

33  m 87 

[ 10*26 

I 6688 1 3 

54 

322019 

9*  80 

990215 

•45 

33 1 803 

10*25 

1 668197 

667582 

53 

I 8 

322607 

323194 

323780 

9-79 

990 1 88 
990161 

990134 

•45 

3324i8 
333o33 
j 333646 

1 10*24 

10*23 
10*21 

52 

5i 

5o 

9 

1 io 

9*77 

0*76 

, *43 

; *45 

1 000007 

| 666354 

n 

9 324366 

9*75 

9*990107 

; -46 

1 9*334259 

j 10*20 

'io*66574i 

49 

13 

i3 

324950 

325534 

9*73 

9*72 

990079 

990052 

.46 
! -46 

334871 

335482 

i 10*19 

I ,0*,7 

665 129 
6645i8 

48 
i 47 

M 

326117 

9-jo 

990025 

! ‘46 

336093 

1016 

663907 

| 46 

1 5 

326700 

9.69 

989997 

•46 

336702 
33731 1 

1 0 - 1 5 

663298 

I 45 

16 

327281 

9*68 

989970 

*46 

io*  i3 

662689 

44 

*7 

327862 

9*66 

989942 

*46 

10*12 

662081 

43 

; id 

328442 

9*65 

989915 

989887 

•46 

10*  1 1 

661473 

42 

i 19 

329021 

9*64 

•46 

339 I 33 

10*  IO 

660867 

41 

30 

329599 

9*62 

989860 

•46 

339739 

10*08 

660261 

4o 

I 3i 

9*330176 

9*61 

9*989832 

•46 

9 • 34o344 

1007 

1 0 • 669656 

39 

i 32 

33o753 

9*  60 

989804 

•46- 

340948 

1006 

659052 

38 

33 

33i32o 
33 1903 

9*58 

989777 

*46 

341552 

1004 

658448 

| 37 

24 

9.57 

989749 

•47 

342i55 

io*o3 

657845 

36 

25 

26 

332478 

333o5i 

9*5o 

9*54 

9*53 

$$ 

•47 

*47 

•47 

342757 

343358 

343958 

10*02 

10*00 

657243 

65o642 

656o42 

1 35 
! 34 

1 33 

2I 

333624 

989665 

9.90 

30 

334195 

9*52 

989637 

•47 

344558 

9*98 

655442 

32 

29 

334766 

335337 

9*  5o 

989609 

•47 

346167 

997 

664848 

3i 

3o 

9.49 

989582 

•47 

345755 

9.96 

654245 

3o 

3i 

9*335906 

9.48 

9*989553 

•47 

9*346353 

9-94 

io*653647 

29 

32 

336475 

337043 

9.46 

9*45 

989625 

989497 

080460 

•47 

•47 

346q4q 

9*93 
9*92  j 

653o5i 

28 

33 

347?45 

652455 

27 

34 

35 

337610 

338i76 

9.44 

9*43 

.47 

•47 

348141 

348735 

9*91 
99°  1 

65i 85o 

26 

989441 

65 1 265 

25 

36 

338742 

9*4i 

989413 

•47 

349829 

9.88 

660671 

24 

339306 

9*4o 

989384 

•47 

349922 

35o5i4 

9*87 

650078 

23 

36 

339871 

340434 

9.39 

989356 

•47 

9.88 

649486 

22 

39 

9-3  7 

0893 28 

•47 

35i 106 

9*85 

640894 

21 

40 

340996 

9*36 

989300 

•47 

351697 

9*83 

6483o3 

20 

41 

9*34i558 

9*35 

9.989271 

•47 

9*352287 

9*82 

io*6477i3 

’9 

42 

3421 19 

9*34 

989243 

•47  1 

352876 

353465 

9 • 8 1 

647124 

l8 

43 

342679 

9*32 

989214 

•47 

9*  80 

646535 

IT 

44  | 

343239 

9 • 3 1 

989186  1 

I -47  1 

354o53 

9-79 

645947  : 

6 

45  1 

344913 

9*3o 

989157  ' 
9891 28 
989 I 00 

1 *47  i 

354640 

977 

645360  1 

|5 

1 46  i 
47 

9.29 

9*27 

■48 ; 
.48 1 

355227 
3558 1 3 

9‘75  | 

644773 
644187  ; 

14 

i3 

I 40 

345469 

9*26 

98907 1 

•48 1 

356398 

9-74  | 

643602  ; 

f 2 

1 49 

346024 

9 25 

989042 

.48 

356982 

9*73  1 

643018 

ri 

1 5o 

346679 

7*24 

0890 1 4 

.48 

357566 

9-7i  I 

642434 

10 

5i 

9*347i34 

5*22 

9*988985 

.48 

9*358i49 

9*70  ! 

io*64i85» 

9 

) 52 

347687 

9.21 

988956 

.48 

368731 

359313 

960 

641269 

8 

1 53 

3 18240 

9*20 

988927 

088808 

.48 

9*68 

640687 

7 

348792 

349343 

• Aft 

359893 

640107 

5 

55 

9*  19 

9*17 

988869 

.48 

360474 

9*  60 

639520 

5 

; 56 

349893 

9*  10  | 

988840 

.48 

36io53  ^ 

9*65 

638947 

638368 

4 

53 

35o443 

9*i5  , 

988811 

.49 

36i632 

9*63 

3 

58 

350992  | 
35 1040  1 

9*!4 

988782 

•49 

362210 

9*62 

03779° 

2 

59 

9*  1 3 

988753 

•49 

362787  I 

9*61 

637213 
636636  | 

1 

60 

352088  | 

911 

988724 

•49 

363364 

q*6o  j 

0 

Coniine  | 

I). 

Sine 

Cotang.  j 

D.  ^ 

Tang.  J 

M. 

(77  DK£KSCtO 


SIXES  AND  TANGENTS.  (13  DEGREES.) 


3) 


M. 

0 

i 

a 

3 

4 

5 

e 

1 

9 

10 

la 

i3 

*14 

15 

16 

12 

>9 

20 

ai 

aa 

23 

24 

25 

26 

2 

ll 

3i 

3a 

33 

34 

35 

36 

39 

40 

41 

42 

43 

44 

45 

46 

2 

«■ 

S Si 
5a 
! 53 

1 54 
! 55 
56 

U 

*9 

60 

Sino 

D. 

Cosine 

D. 

Tang.  | D. 

Cotang. 

.>  35ao88 
352635 
353 i 8i 
353726 
354271 

3548i5 

355358 

355901 

356443 

356984 

357024 

9- 358o64 
3586o3 
359141 
359678 
36oa 1 5 
360752 
361287 
361822 
362356 
362889 

9*363422 

363954 

364485 

365oi6 

355546 

366075 

3666o4 

367i3i 

367659 

368i85 

9*36871 1 
369236 
369761 
370285 
370808 
37i33o 
371852 
372373 
372894 

373414 

9*373933 

374452 

37497° 

375487 

376003 

376519 

377035 

377549 

378063 

378577 

9.379089 
379601 
38oi i3 
380624 
A81 1 34 
38i643 
382152 
38a66i 
383 1 68 
383675 

911 

9- 10 
9-00 

9 08 
9-07 
9*o5 

9”  04 
9*o3 
9*02 
9*01 
8.99 

8*98 

8*95 

8*93 

892 

8*91 

8*90 

8*89 

8*88 

8*87 

8*85 

8*84 

8*83 

8*82 

8*8i 

8*8o 

8*79 

S:?Z 

W 

8*73 

8*72 

8*71 

8*70 

8*69 

8*67 

8*66 

8*65 

8*64 

8*63 

8*62 

8-61 

8*  60 
8*59 
8*58  1 

S:S  1 

9*54 

8*53 
8*52  | 

8 • 5l  ; 
8*5o  ! 
8.49 

8.48 

8*47 

8.46 

8*45 

8-44 

Q 988724 
988696 
988666 
988636 
988607 
988578 
988548 
988519 
988489 
988460 
988430 

9-988401 
988371 
988342 
9883 1 2 
988282 
988252 
988223 
988193 
988163 
988i33 

9*988103 

988073 

988043 

988013 

987983 

987953 

987922 

987892 

987862 

987832 

9*987801 

987771 

987740 

987710 

987679 

987649 

987618 

987688 

987557 

987526 

9*987496 

987465 

987434 

987403 

987372 

98^341 

987310 

987270 

987248 

987217 

9.987186 

987155 

987124 

987002 

987001 

987030 

980998 

986967 

986936 

986904 

.49 

.49 

•49 

.49 

.49 

.49 

.49 

.49 

•49 

.49 

.49 

49 

•49 

*49 

• 5o 

• 5o 

• 5o 

• 5o 

• 5o 

• 5o 
•5o 

•5o 

•5o 

*5o 

*5o 

*5o 

• 5o 
*5o 
*5o 

• 5o 

• 5i 

*5i 

*5i 

• 5i 

• 5i  i 

• 5i 

• 5i 
•5i 

• 5i 

• 5i 

• 5i 

• 5i 
*5i 

• 5i 
5a 

*52  1 
*52  ! 

• 5a  ; 
•52 

• 5a 

• 5a 

• 5a 

• 5a  1 
•52 
*52 
•5a 
•5a 
•5a  j 
•5a 
•5a 

• 5a 

9 • 363364 
363q4o 
364Di5 
365090 
365664 
366237 
3668io 
367382 
367953 

368624 

369094 

9 • 369663 
370232 
370799 

1 

372499 

373CO4 

373620 

374196 

374766 

9.3753I9 

37588i 

376442 

377003 

377563 

378122 

37868i 

379239 

&S8 

9*380910 

381466 

382020 

382575 
383 1 29 
383682 
384234 
384786 
385337 
385888 

9*386438 
386987 
387636 
388084 
38863i  ! 
389178 
389724 
390270 
390816 
391360 

9.391903 

392447 

392989 

393531 

394073 

394614 

395i54 

395694 

396263 

396771 

9-60 

! 

$:S 

9.54 

9-53 

9-5a 

9. 5i 
9*5o 
9.49 
9.48 
9*46 
9-45 
9.44 
9*43 
9*42 
9-41 
9.40 

?:l8 

?:S 

9.34 

9*33 

9*32 

9*3i 

9*3o 

9.20 

9*28 

9*27 

9*26 

9*25 

9*24 

9*23 

9*22 

9*21 

9*20 

?:» 

9*17 

9*  i5 
9-i4 
9*i3 
912 
9*ii 
910 

9.°o 

9. 08 
9*07 
9*06 

9*o5 

9*04 

9*o3 

9*02 

9*01 

o*oo 

b 

10-636636 

636o6o 

635485 

634910 

634336 

633763 

633190 

63a6i8 

632047 

631470 

630906 

io*63o337 

629768 

629201 

628633 

628067 

627501 

626936 

626371 

625807 

625244 

1 o- 62468 1 
6241 19 
6a3558 
622997 
622437 
621878 
621319 
620761 
6aoao3 
619646 

10*619090 

6i8534 

617980 

617425 

616871 

6i63i8 

615766 

6i52i4 

6i4663 

614112 

io*6i356a 

6i3oi3 

612464 

61 1916 
611369 
610822 
610276 
609730 
609186 
608640 

10*608097 
607553 
60701 1 
606469 
605927 
6o538o 
604846 
604306 
603767 
603229 

60 

£ 

& 

55 

54 

53 

5a 

5i 

5o 

3 

2 

45 

44 

43 

42 

41 

4° 

2 

2 

35 

34 

33 

3a 

3i 

3o 

2 

2 

25 

24 

23 

22 

21 

20 

2 

2 

i5 

14 

i3 

12 

11 

10 

8 

1 

5 

4 

3 

2 

1 

0 

' Cosine 

D.  | 

Sine 

1 

Cotang.  | D. 

Tang. 

M. 

<J6 


(76  UKOItKKS.) 


62  (14  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 

Sir.e 

D. 

Conine 

n 

I Tang. 

! D- 

Cotang. 

0 

i 

9-383675 

384182 

8-44 

8-43 

9 • 986904 
986873 

•52 

•53 

1 9'39677' 
397309 

8.96 

8.96 

IO-6o3229 

602691 

2 

384687 

8-42 

986841 

•53 

397846 

8.95 

602 1 54 

3 

385192 

8-4i 

986809 

986778 

•53 

| 398383 

8.94 

601617 

4 

385697 

8-40 

•53 

398919 

399455 

1 8.93 

601081 

5 

386201 

8-39 

8-38 

986746 

•53 

l 8.92 

j 6oo545 

6 

386704 

9867 1 4 

•53 

399990 

400624 

8.91 

600010 

7 

387207 

8-37 

986683 

•53 

8-90 

699476 

6 

387709 

388210 

8-36 

Q8665i 

• 53 

4oio58 

8-89 

598942 

9 

8-35 

986619 

• 53 

401691 

B-88 

598409 

597876 

10 

388711 

8-34 

986587 

•53 

402124 

8-87 

ii 

9 • 3892 1 1 

8-33 

9-986555 

• 53 

9*402666 

8-86 

10-597344 

596813 

12 

3897 1 1 

8-32 

986523 

• 53 

4o3i8 7 
4o37io 

8-85 

i3 

390210 

8«3i 

986491 

986409 

• 53 

8.84 

596282 

14 

390708 

8-3o 

•53 

404249 

8-83 

59575i 

i5 

391206 

8-28 

986427 

986395 

•53 

404778 

8-82 

595222 

16 

391703 

8-27 

•53 

4o53o8 

8- 81 

594692 

*7 

392199 

392695 

8-26 

986363 

•54 

4o5836 

8- 80 

.594164 

10 

8-25 

98633i 

•54 

4o6364 

8.70 

8.78 

593636 

*9 

393191 

393685 

8-24 

986299 

986266 

•54 

406892 

593io8 

20 

8-23 

• 54 

407419 

8.77 

59258i 

21 

9.394179 

394678 

395i66 

8-22 

9-986234 

• 54 

9-407945 

408471 

8-76 

io-592o55 

22 

8-21 

986202 

• 54 

8-75 

59 1 520 

59iooj 

23 

8-20 

986169 

• 54 

408997 

409621 

8-74 

24 

395658 

8- 19 

8- 18 

986137 

• 54 

8-74 

590479 

509955 

25 

396150 

986104 

• 54 

410045 

8.73 

26 

396641 

8-i7 

986072 

• 54 

410669 

8.72 

58943 1 

27 

397132 

8-n 

986039 

• 54 

41 1092 

8.71 

588908 

26 

397621 

398111 

8- 16 

986007 

•54 

411616 

8-70 

588385 

29 

8-i5 

985974 

•54 

412137 

412658 

g-S 

587863 

3o 

398600 

8- 14 

985942 

• 54 

8-68 

587342 

31 

32 

9 • 399088 

399575 

400062 

8- 13 
8-12 

9'® 

• 55 

• 55 

9 • 41 3 1 79 
413699 

8-67 

8-66 

io-58682i 

5863oi 

33 

8- 1 1 

985843 

• 55 

414219 

414738 

8-65 

58578i 

34 

4oo54o 

4oio35 

8-io 

9858ii 

• 55 

8-64 

585262 

35 

8-09 

8-o8 

985778 

• 55 

4i5257 

416775 

8-64 

684743 

36 

401 520 

985745 

• 55 

8-63 

584225 

3I 

402005 

8-07 

985712 

• 55 

416293 

8-62 

583707 

38 

402489 

8-o6 

985679 

• 55 

416810 

8- 61 

583 1 90 

39 

40 

402972 

4o3455 

0000 

0 0 

985646 

9856i3 

• 55 

• 55 

417326 

417842 

8- 60 
8-59 

582674 

582158 

4i 

9-403938 

8-o3 

9-985580 

• 55 

9-4i8358 

8-58 

io- 581642 

42 

404420 

8-02 

985547 

• 55 

418873 

419387 

8-5i 

8-56 

581127 

43 

404901 

4o5j82 

8-oi 

9855 1 4 

• 55 

58o6i3 

44 

8-oo 

985480 

• 55 

419901 

8-55 

58oooo 

579585 

45 

4o5862 

7-99 

985447 

• 55 

42041 5 

8-55 

46 

406341 

7.98 

985414 

• 56 

420927 

8-54 

679073 

578600 

! 47 

406820 

98538o 

• 56 

421440 

8-53 

48 

407299 

985347 

•56 

421952 

8-52 

578048 

49 

407777 

7-95 

9853 1 4 

• 56 

422463 

8-5i 

577537 

677026 

5o 

408264 

7-94 

985280 

• 56 

422974 

8-5o 

5i 

9-408731 

7-94 

9-985247 

986213 

• 56 

9-423484 

8.40 

8.48 

io-5765i6 

52 

409207 

7-93 

• 56 

423993 

4245o3 

576007 

53 

409682 

7.92 

985180 

• 56 

8-48 

$$ 

54 

410157 

7.91 

985146 

• 56 

426011 

8-47 

8.46 

56 

4io632 

7.90 

985ii3 

• 56 

**  425519 

574481 

56 

57 

411106 

411579 

412052 

7.80 

7-88 

986079 

985045 

• 56 

• 56 

426027 

426534 

8-45 

8-44 

573973 

573466 

56 

7.87 

985oi 1 

• 56 

427041 

8-43 

59 

412524 

7-86 

984978 

• 56 

427547 

428062 

8-43 

60 

412996 

7-85 

984944 

• 56 

8-42 

571948 

Coni  »e 

IX  | 

•sine 

Uotang. 

D. 

Tang-._j 

(7ft  DEGREES.) 


wwwwwwwwww  v>mu<u<u<uiu<uiu«uos 

**•  O' — I COO  O — M Coi*.  c*  0*-~J  OOO  O «-  M C_n  O — 1 COO  O — >o  u»Ss  c*&-j  OOO  O to  OJAs  <_*  O — I OOO  o 


SINES  AND  TANGENTS.  (15  DEGREES.; 


83 


M. 

Sind 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9 412996 

7-85 

9 ■ 984944 

•57 

9.428052 

8-42 

10-671948 

60 

i 

4I34J*1 

7.84 

984910 

.57 

428557 

8.41 

571443 

59 

a 

413938 

7-83 

984876 

•57 

429062  j 

8.40 

570938 

58 

3 

414408 

7-83 

984842 

•57 

429566 

8.39 

8-38 

670434 

56oo3o 

57 

4 

414878 

415347 

4i58i5 

7-82 

984808 

984774 

984740 

• 5~i 

430070 

43o573 

431075 

56 

5 

6 

7-8! 

7-80 

I7 

•57 

8-38 

8.3? 

569427 

568926 

55 

54 

7 

416283 

7-79 

984706 

•57 

43 1 577 

3-36 

568423 

53 

8 

9 

416751 

417217 

7.78 

7-77 

984672 

984637 

I7 

•57 

432079 

432580 

8-35 

8-34 

567921 

667420 

52 

5i 

10 

417684 

7.76 

984603 

-57 

433o8o 

8-33 

566920 

5o 

ii 

9-4i8i5o 

7.75 

9 • 984569 
984535 

•57 

9 -43358o 

8-32 

10-666420 

49 

12 

4i86i5 

7-74 

• 57 

434080 

8-32 

666920 

48 

»3 

419079 

7-73 

984500 

•57 

434579 

8-3i 

565421 

47 

14 

419^44 

7-73 

9844^6 

•57 

435078  ' 

8»3o 

564922 

46 

i5 

420007 

7-72 

984432 

• 58 

435576 

8-29 

564424 

45 

16 

420470 

7-71 

984397 

-58 

436073 

8-28 

663927 

44 

*7 

- 420933 

7-7° 

984363 

-58 

436570 

8.28 

56343o 

43 

18 

19 

42i3o5 

421807 

7-60 

7-68 

984328 

984294 

-58 
• 58 

437067 

437564 

438o5g 

8-27 

8-26 

562933 

562437 

42 

41 

20 

4223i8 

7-67 

984269 

• 58 

8-25 

561941 

40 

21 

9-422778 

7-67 

9-984224 

-58 

9-438554 

8-24 

10-661446 

3q 

22 

423238 

7-66 

984 I QO 

• 58 

439048 

8-23 

560952 

38 

23 

423697 

7-65 

984165  I 

• 58 

439643 

8-23 

56o457 

37 

U 

424106 

7-64 

984120  j 

• 58 

44oo36 

8-22 

559964 

36 

25 

424615 

7-63 

984085 

-58 

440529 

8-21 

55947 1 

35 

26 

27 

426073 

42553o 

7-62 

7*6i 

984050 

984015 

• 58 

• 58 

441022 

44i5i4 

8-20 

8*  19 

558978 

558486 

34 

33 

28 

425987 

7-60 

983981 

-58 

442006 

8‘i9 

557994 

557603 

32 

19 

426448 

7-60 

983946 

-58 

442407 

8*  18 

3i 

3o 

426899 

7-59 

983911  ! 

• 58 

442988 

8-17 

557012 

3o 

3 1 

9-427354 

7-58 

9-983875 

• 58 

9-443479 

8.16 

io-55652i 

29 

32 

427809 

428263 

7-57 

983840  1 

• 59 

443968 

8*  16 

556o32 

28 

33 

7-56 

9838o5  I 

• 59 

444458 

8-i5 

555542 

27 

34 

428717 

7-55 

983770  I 

-59 

444947 

445435 

8-14 

555o53 

26 

35 

429170 

7-54 

983735  | 

-59 

8-i3 

554565 

25 

36 

429623 

7-53 

983700 

-59 

445923 

8-12 

554077 

553589 

24 

h 

430075 

7-52 

983664 

• 59 

446411  1 

8-12 

23 

38 

43o52-7 

7-52 

983629 

-59 

446898 

8-11 

553io2 

22 

39 

430978 

7 * 5 1 

983594 

•59 

447384 

8-io 

5526i6 

21 

40 

431429 

7-5o 

983558 

•59 

447870 

8-09 

552 1 3o 

20 

41 

9-431879 

7-49 

9-983623 

• 59 

9-448356 

8-09 

io-55i644 

42 

432320 

7-49 

983487 

59 

448841 

8- 08 

55r 1 59 

l8 

43 

432778 

7.48 

983452  ; 

•59 

449326 

8-07 

550674 

17 

44 

433226 

7-47 

983416 

-59 

449810 

8-06 

550190 

l6 

45 

433675 

7.46 

983381 

-59 

460294 

8-  06 

549706 

i5 

46 

| 434122 

j 7-45 

983345 

• 5g 

460777 

8-o5 

549223 

14 

4Z 

| 434569 

7-44 

983309 

• 59 

461260  1 

8-04 

548740 

i3 

435oi6 

I 7.44 

983276 

•60 

451743 

8-o3 

548257 

12 

i 49 

435462 

1 7-43 

983238 

60 

452225 

8-02 

547776 

11 

! 5o 

435908 

7.42 

983202 

•60 

452706 

8-02 

547294 

10 

1 Ji 

! 9-436353 

7.41 

| 9-983166 

•60 

9 • 453 1 87 

8-oi 

io-5468i3 

9 

1 

, 436-/98 

7-40 

983 1 3o 

•60 

453668 

8-oo 

546332 

8 

1 53 

43i242 

1 7-4o 

983094 

•60 

454i4f 

7-99 

545852 

7 

54 

! 437686 

7-3o 

983o58 

-60 

454628 

7-99 

7-98 

545372 

0 

55 

1 438129 

J 7-38 

983022 

-60 

466107 

544893 

5 

56 

438572 

i 7-37 

982986 

•60 

455586 

7-97 

544414 

4 

439014 

439456 

1 7-36 

982960 

•60 

456064 

7.96 

543936 

3 

58 

*9 

7-36 

982914 

-60 

456542 

7.96 

543458 

2 

439897 

44o338 

7-35 

982878 

60 

457019 

7-95 

542981 

1 

1 60 

| 

7-34 

982842 

•60 

457496 

794 

542604 

0 

l 

Coni  no 

D. 

Sino 

Cotang. 

I D. 

Tang. 

M. 

(74  DEOKBRR.) 


(l(i  DEGREES.)  A TABLE  OF  LOGARITHMIC 


S4 


M. 

Sino 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9*44o338 

7-34 

9 982842 

• 60 

9-457496 

7-94 

10 • 542504 

60 

i 

440778 

7-33 

982805 

• 60 

457973 

458449 

458925 

7 -93 

542027 

5q 

2 

441318 

7-32 

982769 

982733 

Q82696 

• 61 

7*93 

54i 55i 

58 

3 

441658 

vil 

• 61 

7-92 

541075 

57 

4 

442096 

442535 

7 * 3 1 

•61 

459400 

7-91 

O40O0U 

So 

5 

7*3o 

982660 

•61 

459876 

7.90 

540125 

, 55 

6 

442973 

?:3 

982624 

•61 

460349 

460823 

7.00 

539651 

1 54 

7 

443410 

982687 

•61 

7.80 

7-88 

j 53 

1 8 

443847 

7-27 

982551 

•61 

461297 

1 52 

9 

444284 

7-31 

7*26 

982514 

•61 

461770 

7-88 

53823o 

5i 

10 

444720 

982477 

•61 

462342 

7-87 

537758 

5o 

ii 

9-445i55 

1*25 

9-982441 

•61 

9-462714 

7.86 

10-537286 
5368 1 4 

49 

13 

445590 

7-24 

982404 

•61 

463 1 86 

7-85 

48 

i3 

446035 

7*23 

982367 

•61 

463658 

7-85 

536342 

47 

14 

446459 

446893 

447336 

7*23 

982331 

•61 

464129 

7*84 

53587i 

46 

15 

16 

7*22 

7*21 

982294 

982207 

•61 

•61 

464599 

465069 

7-83 

7-83 

5354oi 

53493i 

45 

44 

*7 

447759 

448191 

7-20 

982220 

•62 

465539 

466008 

7-82 

534461 

43 

1 8 

7*20 

982183 

•62 

7*8i 

533992 

533524 

42 

'9 

448623 

982146 

•62 

466476 

7-80 

4i 

30 

449064 

982109 

•62 

466945 

7-80 

533o55 

4o 

31 

9*449485 

7-,7 

9-982072 

•62 

9-467413 

7-79 

10-532687 

3g 

33 

449915 

45o345 

7*i6 

982o35 

•62 

467880 

7.78 

532120 

38 

33 

7. 16 

981998 

•62 

468347 

7.78 

53 1 653 

37 

34 

460775 

7 * * 5 

981961 

•62 

468814 

I'll 

53i 186 

36 

35 

45 1 204 

7*14 

981924 

981886 

•62 

469280 

7.76 

530720 

35 

36 

45 1 632 

•62 

An 

469746 
4702 1 1 

T7* 

7-75 

530254 

34 

O') 

981849 

27 

452o6o 

7*13 

• 02 

529324 

528869 

528395 

J J 

36 

453488 

7-13 

981812 

•63 

470676 

7-74 

32 

39 

3o 

452915 

453342 

7' 11 
710 

981774 

981737 

•62 

•62 

47»i4i 

47i6o5 

7.73 

7.73 

Hi 

3i 

9-453768 

7*io 

9-981699 

981662 

•63 

9-472068 

7-72 

10-627932 

3 

33 

454194 

?:3 

•63 

472532 

7*71 

527468 

33 

454619 

981625 

•63 

472995 

473457 

TV 

527005 

a7 

34 

455o44 

7*07 

981587 

•63 

VI 0 

526543 

26 

35 

36 

455469 

455893 

?:3 

981549 
981 5 1 2 

•63 

•63 

473919 

474381 

7-69 

7-60 

526081 

626619 

25 

24 

ll 

4563 1 6 

7*o5 

981474 

981436 

•63 

474842 

7-68 

6261 58 

23 

466739 

7-04 

•63 

4753o3 

7-67 

524697 

22 

39 

457162 

7*o4 

981399 

98i36i 

•63 

475763 

7.67 

524237 

21 

40 

457584 

7*o3 

•63 

476223 

7-66 

523777 

20 

U 

9 • 458oo6 

7.02 

9-981323 

• 63 

9-476683 

7-65 

10-623317 

» 

43 

458437 

7*oi 

981285 

•63 

477142 

7-65 

522858 

43 

458848 

7.01 

981247 

•63 

477601 

478059 

7-64 

522399 

>7 

44 

469268 

7-  00 

981209 

•63 

7-63 

521941 

16 

45 

459688 

6*90 

6-98 

981171 

•63 

478517 

7-63 

521483 

i5 

46 

460108 

981 i33 

• 64 

478975 

479432 

7-62 

521025 

14 

460621 

6*98 

981095 

981007 

981019 

•64 

7.61 

52o568 

i3 

ii 

1 49 

460940 

461364 

•64 

•64 

479889 

4oo34o 

7-6! 

7-60 

520111 

5 1 9655 

12 

11 

5o 

461783 

6*95 

980981 

•64 

480801 

7-59 

519199 

10 

5i 

9 462199 
462616 

6 95 

9-980942 

• 64 

9-481257 

7-5o 

io-5i8743 

1 

52 

6*94 

980904 

980866 

• 64 

481712 

7-58 

518288 

53 

463o32 

6*93 

•64 

482167 

7-5? 

5 i7833 

7 

54 

463448 

6-93 

980827 

• 64 

482621 

7-57 

i:g3 

0 

55 

463864 

6*92 

980789 

•64 

483075 

7-56 

5 

56 

464279 

6*91 

980750 

•64 

483529 

7 • 55 

516471 

4 

57 

464694 

6 90 

9801 1 2 
980673 
980635 

•64 

483982 

7-55 

5i6oi8 

3 

58 

465 1 08 

690 

• 64 

484435 

7-54 

5 1 5565 

2 i 

59 

465533 

6.89 

•64 

484887 

7-53 

5i5ii3 

1 , 

60 

465935 

6*88 

980596 

•64 

485339 

7-53 

614661 

•J 

CoHino 

D. 

Si  no 

Cotang. 

D. 

Tang.  1 

i M.  J 

(73  DRGHRR8.) 


SINKS  AND  TANGENTS.  (17  DEGREES.) 


3fc 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

i 

9-465o35 

466348 

6-88 

6-88 

9 • 980596 
98o558 

•64 

.64 

9485339 

485791 

7-55 

7-52 

io-5i466i 

514209 

60 

5o 

2 

466761 

6-87 

980519 

-65 

486242 

7 * 5i 

5i3758 

58 

3 

467173 
467 585 

6-86 

980480 

•65 

486693 

7 * 5i 

5 1 3307 

57 

4 

6-85 

980442 

•65 

487143 

7*5o 

512857 

5 

467996 

6-85 

980403 

•65 

487593 

488043 

7*49 

612407 

55  ; 

6 

468407 

6-84 

980364 

-65 

7-4  9 

5i 1067 
5ii5o8 

54 

7 

468817 

6-83 

980325 

•65 

488492 

7-48 

53 

8 

469227 

6-83 

980286 

•65 

488941 

7-47 

5i  io:>9 

52 

9 

10 

46963 7 
470046 

6-82 

6-8i 

980247 

980208 

•65 

-65 

489390 

489838 

7-47 

7.46 

5io6io 

510162 

5i 

5o 

»« 

9-470455 

6-8o 

9-980169 

-65 

9-490286 

7.46 

10-509714 

49 

12 

470863 

6- 80 

980130 

-65 

490733 

7-45 

600267 

608820 

48 

i3 

471271 

6-79 

980091 

980062 

• 65 

491 180 

7-44 

4] 

14 

471679 

472086 

6-78 

-65 

491627 

7-44 

5o8373 

46 

i5 

6-78 

980012 

-65 

492073 

7-43 

507927 

45 

16 

ll 

472492 

472898 

979973 

979934 

979895 

•65 
• 66 

492619 

492965 

7-43 

7*42 

507481 

607035 

44 

43 

18 

4733o4 

6-76 

• 66 

493410 

7.41 

506590 

42 

*9 

473710 

6-75 

979855 

• 66 

493854 

7.40 

5o6i46 

41 

20 

474i i5 

6-74 

979816 

• 66 

494299 

7.40 

5o57oi 

4o 

21 

9-4745iq 

6-74 

9-979776 

• 66 

9-494743 

7.40 

io-5o5257 

32 

22 

474920 

6-o3 

979737 

■ 66 

495186 

7-3o 

604814 

38 

23 

475327 

6-72 

979697  | 

•66 

49563 0 

7*38 

504370 

37 

*4 

475730 

6-72 

979658 

•66 

496073 

7.37 

503927 

36 

25 

476i33 

6-71 

979618 

• 66 

4965i5 

7-37 

5o3485 

35 

26 

3 

29 

476536 

476o38 

477040 

477741 

478142 

6-70 

6-09 

979579 

979539 

• 66 
•66 

496067 

497399 

7.36 

7-36 

5o3o43 

5o26oi 

34 

33 

6-69 

6-68 

979499 

979459 

•66 

•66 

497841 

498282 

7*35 

7-34 

5o2 1 5o 
501718 

32 

3i 

3o 

6-67 

979420 

•66 

498722 

7-34 

601278 

3o 

3i 

9-478542 

6-67 

9-979380 

• 66 

9-499163 

7 33 

10 -600837 

39 

32 

478042 

6-66 

979340 

•66 

499603 

7-33 

5oo3o7 

499938 

499319 

28 

33 

479042 

6-65 

979300 

• 67 

600042 

7-32 

2J 

34 

479741 

480140 

6-65 

979260 

.67 

600481 

7 * 3 1 

26 

35 

6-64 

979220 

•67 

500020 

601359 

7*3i 

499080 

498641 

2t> 

36 

480539 

6-63 

979180 

.67 

7-3o 

24 

ll 

480937 

481334 

6-63 

6-6i 

979140 

979100 

‘a7 

67 

&& 

7-3o 

7-29 

498203 

497765 

497328 

496891 

23 

22 

39 

481731 

6 • 6 1 

979069 

•67 

502672 

7-28 

21 

40 

482128 

6-6i 

979019 

• 67 

5o3io9 

708 

20 

4i 

9-482525 

6-  60 

9.978979 

• 67 

9-5o3546 

7.27 

10  496454 

19 

42 

43 

482921 

4833i6 

6-59 

6^ 

978930 

978898 

.67 

•67 

603982 

5o44i8 

7'27 

7-26 

496018 

495582 

10 

I] 

44 

483712 

978868 

.67 

5o4854 

7-25 

495146 

l6 

45 

484107 

1 6-67 

978817 

.67 

505289 

7*25 

494711 

i5 

46 

4845oi 

! 6-57 

$£ 
am  q/LaA 

• 67 

506724 

7-24 

494276 

14  ! 

s 

484895 

6-56 

A AA 

5o6i5o 

5o6593 

7.24 

7-23 

493841 

493407 

492973 

492640 

i3  | 

1 11 

40 

40D209 

O • JJ 

C)^oOQO 

• DO 

1 3 

49 

485682 

6-55 

978665 

•68 

507027 

7*22 

1 11 

5o 

| 486075 

1 6-54 

978615 

• 68 

507460 

| 7-22 

10 

5i 

1 9-486467 

6-53 

9-978574 

978533 

•68 

9007893 

5o8326 

| 7*21 

10-492107 

9 

52 

486860 

6-53 

• 68 

7-21 

491674 

8 

53 

487251 

6-52 

978493 

• 68 

508759 

7-20 

491241 

7 

54 

487643 

6 - 5 1 

978452 

• 68 

509191 

1 719 

490809 
49037 8 

0 

55 

488034 

6-5i 

978411 

• 68 

509622 

7. 10 

5 

56 

488424 

6-5o 

978370 

• 68 

5 10064 

7-18 

489946 

4 

bl 

488814 

6-5o 

978329 

978288 

978247 

68 

5io485 

7.18 

489515 

3 

58 

59 

489204 

489593 

489982 

6-40 

6-48 

• 68 
• 68 

5iooi6 
5i 1 346 

?:a 

489084 

488654 

2 

60 

6-48 

978206 

• 68 

5i 1776 

716 

488224 

0 

Coeine 

i D. 

i Sine 

D. 

Cotang. 

1 D. 

Tang. 

M. 

17 


(72  DEGREES.) 


O y£  a>-J  O'U'MJIO  - fl<l  CfrJO>Uii.WM-  Ov6 


S6 


(18  DEGKEE8.)  A TABLE  OF  LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

2 

3 

4 

5 

6 

l 

9 

10 

u 

13 

13 

14 

15 

16 

:z 

'9 

30 

31 

33 

33 

34 

35 

36 

3 

11 

3i 

33 

33 

34 

35 

36 

U 

39 

40 

41 
43 

43 

44 

45 

46 

s 

s 

5i 

5s 

53 

54 

55 

56 

U 

s 

9-489083 
490 J71 
490769 

491147 
491535 
491933 
493308 
493605 
493081 
493466 
49385 1 

9-494336 

494621 

495oo5 

495388 

495773 

496164 

496537 

496919 

497501 

497683 

9 • 498064 

498444 

498825 

499204 

499584 

499963 

5oo342 

500721 

501099 

501476 

9-5oi854 
5o223l 
602607 
602984 
5o336o 
5o3735 
5o4i 10 
5o4485 
604860 
5o5234 

9 • 5o56o8 
605981 
5o6354 
506727  | 
507099 
507471 
507843 
5oo2 14 
5o8585 
5o8956 

9.509326 

609696 

5ioo65 

610434 

5io8o3 
511173 
5ii54o 
5i 1901 
512275 
513643 

6-48 

6-48 

6-47 

6-46 

6-46 

6-45 

6-44 

6-44 

6-43 

6-43 

6-42 

641 

6-41 

6-40 

6-39 

6-39 

6-38 

6-37 

6*37 

6-36 

6-36 

6-35 

6-34 

6-34 

6-33 

6-32 

6-32 

6 • 3 1 
63i 
6-3o 
6-29 

6-29 

6.28 

6-28 

6.27 

6-26 

6-26 

6-25 

6-25 

6-24 

6-23 

6-23 

6-22 

6-22 

6-21 

6-20 

6-20 

6-19 

6-19 

6-l8 

6-l8 

£3 

6. 16 

6 * ! 5 

6- 15 

6- 14 

6 • 1 3 

6- 13 
6-12 
6*12 

9-978206 

978165 

978124 

978083 

978042 

978001 

977950 

977918 

sbSb 

977794 

9.977752 

977628 

977686 

977544 

9775o3 

97746i 

977419 

977377 

9-977335 

977203 

97725i 

977209 

977167 

977125 

977083 

977041 

976909 

976967 

9.976014 

976872 

9768J0 

976787 

976745 

976702 

976060 

976617 

976674 

976532 

9.976489 

976446 

976404 

976361 

676318 

976275 

976252 

976189 

976146 

976103 

9-976060 

976017 

975974 

975930 

975887 

976844 

975800 

975757 

975714 

975670 

• 68 
•68 
• 68 

■J9 

*J9 

.69 

■69 

• 69 
-69 
-69 
•69 
•70 
.70 
.70 
•70 
.70 

.70 

.70 

.70 

.70 

.70 

•70 

.70 

.70 

.70 

.70 

.70 

•7i 

.71 

•7i 

•7i 

•7i 

•7i 

•7i 

•7i 

•7i 

•7i 

•7i 

•7i 

•71 

.71 

•7i 

.72 

.72 

•72 

.72 

.72 

.72 

•72 

•72 

•72 

•72 

.72 

•72 

.72 

.72 

9 511776 
512206 
512635 
5i3o64 
5i3493 
5i3o2I 

514549 

514777 

5i52o4 

5i563i 

516057 

9.516484 

516910 

617535 

617761 

5ioi85 

5i86io 

519034 

519468 

519882 

52o3o5 

9-620728 
521 1 5i 
521573 
621995 
622417 
522838 
523259 
52368o 
524100 
524520 

9.524939 

525359 

526778 

526197 

5266i5 

527033 

527461 

527868 

528286 

528702 

9-629119 

529535 

629950 

53o366 

530781 

531196 

53i6ii 

532025 

532439 

532853 

9-533266 
533679 
534093 
5345o4 
534916 
535328 
535739 
536 1 5o 
53656i 
536972 

7.16 

7.16 

7**5 

7-14 

7-14 

7**3 

7 • i3 
7.12 
7.12 
7.11 
7.10 

7-io 

7-09 

7-oo 

7-08 

7- 08 

7-06 

7-o5 

7-o5 

7-04 

7*o3 

7-o3 

7-o3 

7-02 

7-02 

7-oi 

7-oi 

1-00 

6.99 

6# 

6.98 

A*97 

6# 

6.96 

6.95 

6.95 

6-94 

6.g3 

6.63 

693 

6.62 

6.91 

6.61 

6.60 

6.60 

6.89 

6.89 

6.88 

6-88 

6-87 

6.87 

6-86 

6-86 

6-85 

6-85 

6-84 

6-84 

10. 488224 
487794 
487365 
486936 
j 486607 
l 486079 
48565 1 
485223 
484796 
484469 
483944 
io-4835i6 
483090 
482665 
482239 
481816 
481390 
480906 
480643 
480118 

47969s 

10-470272 

478849 

478427 

478006 

477583 

477162 

476741 

476320 

475900 

475480 

10 -475o6i 

474641 

474222 

4738o3 

473385 

472967 

472549 

472132 

471715 

471298 

10-470881 

470465 

47oo5o 

469634 

460219 

468804 

46838o 

467076 

467661 

467147 

10-466734 

466321 

466908 

466496 

465084 

464672 

464261 

463850 

463439 

463028 

60 

£ 

57 

56  j 
55  1 

54  ! 
53  ! 
53 

5i 

5o 

3 

3 

45 

44 

43 

43 

4i 

4o 

& 

35 

34 

33 

3a 

3i 

3o 

3 

3 

25 

24 

23 

22 

21 

20 

:g 

3 

i5 

14 

i3 

12 

1 1 

.0 

§ 

5 

4 

3 

1 

1 

0 

Coeiuo 

D. 

Sine 

D. 

Cotang. 

D. 

lang. 

M.  j 

(71  DKUKKKB.) 


SINES  AND  1ANGENTS.  (19  DEGREES.) 


87 


M. 

Sine 

D. 

Cosine 

D. 

1 Tang. 

D. 

Cotang. 

o 

9-512642 

6-12 

9.976670 

.73 

9-536972 

6-84 

io-463o:  8 

60 

i 

5i3oo9 

5i337D 

6- 1 1 

975627 

.73 

537082 

6-83 

462618 

59 

a 

6- 11 

q75583 

•73 

537792 

538202 

6-83 

462208 

58 

3 

5i374i 

610 

975539 

•7j 

6-82 

461798 

461389 

5? 

4 

514107 

6-09 

976496 

975402 

•73 

5386 1 1 

6-82 

56 

5 

514472 

6-09 

6- 08 

.73 

539020 

6*8i 

460980 

55 

6 

514837 

976408 

•73 

539429 

6*8i 

460671 

54 

! i 

5i5202 

6.08 

975365 

•73 

53983i 

540245 

6-8o 

460163 

53 

8 

5 1 5566 

6*07 

975321 

•73 

6- 80 

459755 

469347 

458939 

5a 

10 

5i593o 

516294 

6*07 

6-o6 

975a33 

•73 

.73 

54o653 

541061 

6-79 

6.79 

5i 

5o 

II 

9>5i6657 

6-o5 

9.976189 

•7J 

9.541468 

6-78 

10*458532 

49 

12 

517020 

6-o5 

975i45 

.73 

541875 

542281 

6-78 

458125 

48 

i3 

517382 

6-04 

976101 

•73 

6-77 

457719 

47 

14 

517745 

518107 

6-04 

975067 

975oi3 

.73 

542688 

467312 

46 

i5 

6-o3 

.73 

543094 

466906 

45 

16 

518468 

6-o3 

974969 

974925 

974880 

•74 

543499 

543oo5 

544010 

6-76 

4566oi 

44 

17 

518829 

6-02 

•74 

6*75 

456095 

43 

i8 

519100 

6-oi 

•74 

6*75 

455690 

455285 

42 

19 

519551 

6-oi 

974836 

•74 

544715 

6*74 

4i 

20 

519911 

6-oo 

974792 

•74 

545119 

6*74 

454881 

4o 

21 

9*520271 

6-oo 

9'974748 

•74 

9 545524 

6-73 

IO-454476 

39 

22 

5ao63i 

5-99 

974703 

•74 

545928 
54633 1 

6-73 

454072 

38 

23 

520990 

521849 

5-99 

5-98 

974659 

•74 

6*72 

453669 

37 

24 

974614 

•74 

546735 

6-72 

453266 

36 

25 

521707 

5.98 

974570 

•74 

547138 

6-71 

452862 

35 

26 

522066 

5.97 

974525 

•74 

547540 

6-71 

45246o 

34 

27 

522424 

5.96 

974481 

•74 

547943 

548045 

6*70 

452057 

33 

26 

622781 

5-96 

974436 

•74 

6*70 

45 1 656 

32 

29 

523i38 

5-95 

974391 

•74 

548747 

6-69 

45 1 253 

3i 

3o 

523495 

5-95 

974347 

•75 

549149 

6*69 

45o85i 

3o 

3i 

9-523852 

5.94 

9 -9743o2 

.75 

9-54955o 

6-68 

io-45o45o 

29 

3a 

33 

524208 

524564 

5-94 

5-93 

974257 

974212 

.75 

•75 

549961 

55o352 

6-68 

6*67 

460049 

449648 

28 

27 

34 

524920 

6-93 

974167 

•7j 

55o752 

6-67 

6-66 

449248 

26 

, 35 

525275 

6-92 

974122 

.75 

55n5a 

448848 

25 

36 

52563o 

5-91 

974077 

.75 

55i552 

6-66 

448448 

24 

37 

525984 

52633a 

526693 

5-91 

974032 

.75 

551962 

55235i 

6-65 

448048 

23 

38 

5-90 

973987 

.75 

6-65 

447649 

22 

39 

5-90 

5-89 

973942 

973891 

.75 

552760 

6-65 

44725o 

21 

40 

527046 

.75 

553 1 49 

6-64 

44685i 

20 

41 

9-627400 

5-89 

5-88 

9*973852 

•75 

9.553548 

6-64 

io-446452 

:s 

42 

527753 

528105 

973807 

• 75 

553946 

554044 

6-63 

446o54 

43 

5-88 

973761 

•75 

6-63 

445656 

n 

44 

528458 

5.87 

5-87 

973716 

.76 

554741 

6.62 

445269 

16 

45 

528810 

973671 

.76 

555 1 39 

6-62 

444861 

i5 

46 

529161 

5-86 

973625 

.76 

555536 

6- 61 

444464 

14 

47 

5295i3 

5-86 

97358o 

.76 

555933 

556029 

55672S 

6-6i 

444067 

i3 

; 48 

629864 

5-85 

973535 

•76 

6-6o 

443671 

12 

i 49 

; 53o2i5 

5-85 

973489 

•76 

6.60 

443275 

11 

5o 

53o565 

5-84 

973444 

.76 

557121 

6.59 

442879 

10 

5i 

| 9 530916 

5-84 

9.973398 

973352 

.76 

9.557517 

6-59 

10-442483 

J 

5j 

531265 

5-83 

.76 

557910 

6-59 

6-58 

442087 

53 

53i6i4 

5.82 

973307 

.76 

5583o8 

441692 

7 

54 

53io63 

5-82 

973261 

•76 

558702 

6-58 

441298 

0 

55 

532312 

5- 81 

9732i5 

•76 

559097 

6*57 

440903 

5 

56 

53a66i 

5- 81 

973169 

.76 

559491 

559885 

6*57 

6-56 

440609 

4 

57 

533oo9 

5.8o 

973124 

•76 

440116 

3 

58 

59 

533357 

533704 

5.8o 

5$ 

973078 

973002 

.76 

•77 

560279 
56o6i0 
56 1 066 

6-56 

6-55 

439721 

439827 

2 

1 

60 

534o5a 

972986 

•77 

6-55 

438934 

0 

1 

L 

Cosine 

D. 

Sine 

D.  1 

Cotang. 

D. 

Tang  1 

l M. 

(70  DKOKEK8.) 


(20  DKGREEbJ  A TABLE  OF  LOGARITHMS 


38 


M. 

Slue 

I). 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9-534o52 

5.78 

9*972986 

•77 

9*56io66 

6*55 

10-438934 

— 

60 

i 

534399 

5*77 

972940 

•77 

561459 

6*54 

438$4i 

59 

a 

534745 

5*77 

972894 

•77 

56i 85 1 

6*54 

438149 

58  1 

3 

4 

535092 
• 536438 

5*77 

5-76 

972848 

972802 

•77 

•77 

562244 

562636 

6-53 

6*53 

1 437756 

43ij64 

3| 

5 

535i83 

•5*76 

972755 

77 

563028 

6*53 

436972 

436$8i 

55 

6 

536129 

5*75 

972709 

972663 

77 

563419 

6*52 

54 1 

1 

536474 

5*74 

•77 

5638 1 1 

6*52 

436189 

53 

1 6 

5368 18 

5*74 

972617 

•77 

564202 

6*5i 

435798 

5a 

! 9 

537i63 

5*73 

972570 

•77 

564592 

6*  5i 

4354o8 

5i 

10 

537507 

5 - 73 

972524 

•77 

564903 

6*5o 

435oi7 

5o  ! 

n 

9-53785i 

5*72 

9.972478 

97243i 

•77 

! 9*565373 

6*5o 

10*434627 

49 

12 

538194 

5*72 

.78 

‘ 565763 

6*49 

434237 

48  ; 

i3 

538538 

5*71 

972385 

.78 

566 1 53 

6*49 

433847 

433458 

47 

14 

53888o 

5*71 

972338 

.78 

566542 

6.49 

46 

i5 

539223 

5*70 

972291 

.78 

566o32 

6*48 

433o68 

45 

16 

539565 

5*70 

972245 

.78 

567520 

6*48 

432680 

44 

17 

lB 

539907 

540249 

5*69 

972198 
972 1 5i 

.78 

.78 

567709 

568o98 

6*47 

6*47 

432291 
43 1 902 

43 

42 

5*09 

19 

54o5qo 

5*68 

972105 

.78 

568486 

6*46 

43 1 5i 4 

41 

20 

540931 

5*68 

972o58 

.78 

568873 

6*46 

431127 

40 

21 

9*541272 

5*67 

9*97201 1 

•7» 

9*569261 

6*45 

10 *43oi39 
43o$5a 

32 

22 

54161 3 

5*67 

5*66 

971964 

• 78 

569648 

6*45 

38 

23 

541953 

97*917 

*7« 

570035 

6*45 

429965 

429678 

37 

24 

542 2o3 
542632 

542971 

5*66 

971870 

.78 

570422 

6*44 

36 

*5 

26 

5*65 

5*65 

971823 

971776 

.78 

.78 

570809 
j 57119$ 

6*44 

6*43 

429191 

428805 

35 

34 

27 

5433 10 

5*64 

971729 

•79 

j 571 58i 

6*43 

428419 

33 

28 

543649 

5*64 

971682 

•79 

1 671067 

6*42 

428o3j 

32 

29 

543q8] 

5*63 

971635 

•79 

: 572352 

6*42 

427648 

3i 

3o 

54432$ 

5*63 

971588 

•79 

572738 

6*42 

427262 

3o 

3 1 

9*544663 

5*62 

9*971540 

•79 

9*573i23 

6*41 

10*426877 

42649J 

29 

3*3 

54$ooo 

5*62 

971493 

•79 

573507 

6*41 

28 

33 

545338 

5-6i 

971446 

•79  1 

573892 

6*40 

426108 

27 

34 

35 

545674 

546011 

5* 61 
5*6o 

971398 

971351 

•79  | 
•79 

574216 

574660 

6*40 

6*39 

425724 

425340 

26 

25 

36 

37 

546347 

546683 

5*  60 
5*59 

97i3o3 

971256 

•79 

•79 

576044 

576427 

6*39 

6*3o 

424966 

424673 

24 

23 

38 

547019 

5.59 

971208 

•79 

576810 

6*38 

424190 

22 

39 

547354 

5*58 

971161 

•79 

576193 

6-38 

423807 

21 

40 

547689 

5*58 

97Hi3 

•79 

576576 

6*37 

423424 

20 

41 

9.548024 

5*57 

9*971066 

*8o 

9*576958 

677341 

6*37 

io*423o4i 

‘9 

42 

548359 

548693 

5.57 

971018 

• 80 

6*36 

422669 

l8 

43 

5*56 

970970 

• 80 

577723 

6*36  1 

422277 

*7 

44 

549027 

5*56  | 

970925 

• 80  I 

578104 

6*36  ; 

421896 

16 

45 

549360 

5*55  1 

970874 

80 

578486 

6*35  ! 

42 1 5i4 

5 

46 

549693 

5*55  i 

970827 

• 80 

578867 

6*35  j 

421 i33 

U | 

47  j 

55oo26 

5*54 

970779 

97073 1 

•«°  | 

579248 

6*34  | 

420752 

13  ; 

48  1 

55o359 

5*54 

• 80 

579629 

580009 

6*34  1 

42087 1 

1 2 1 

49 

550692 

5*53 

970683 

• 80 

6*34 

419991 

1 1 1 

60 

55io24 

5*53 

970635 

• 80 

58o389 

6*33  | 

419611 

to 

5i 

9*55i356 

5*52 

9 *970586 

• 80 

9*580769 

6*33  ; 

10  419231 
4i885i 

9 

5a 

55i687 

552oi8 

5*52 

970538 

*8o 

58i 149 

6*32  I 

8 

1 53 

5*52 

970490 

• 80 

58i528 

6*32  1 

418472 

7 

I 54 

552349 

5*5i  j 

970442 

*8o 

581907 

582286 

6*32 

418093 

6 

! 55 

55268o 

5*5i  1 

970394 

• 80 

6 • 3 1 

417714 

5 

j 56 

553oio 

5*5o  I 

970346 

-81 

582665 

6 • 3 1 

417836 

4 

1 

55334i 

5*5o 

970297 

• 81 

583043 

6*3o 

416957 

416678 

3 

l 58 

553670 

5*49  ! 

97^249 

•81 

583422 

6*3o 

2 

: 5o 

554ooo 

5*49 

770200 

•81 

5838oo 

6*29 

416200 

1 

60 

554329  j 

5*48 

970.52 

• 81 

584177 

6*29  1 

415823 

0 

Oowino  1 

L>.  1 

Blue 

1>.  ! 

Cotang.  ! 

7).  | Tung. 

M. 

(6\)  UKOItKSH.) 


SINES  AJND  TANGENTS.  (21  DEGREES.) 


39 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

I 

0 

i 

9-554320 

554658 

5-48 

5-48 

9-970152 

970103 

• 81 

• 81 

9.584177 

584555 

6-29 

6-29 

io- 41 5823 
4i5445 

60 

5q 

a 

3 

554987 
5553 1 5 

5-47 

5-47 

970055 

970006 

■ 81 
•81 

584932 

5853o9 

6-28 

6-28 

4i5o68 

414691 

58 

57 

4 

555643 

5-46 

969957 

•81 

585686  ’ 

6-27 

4i43i4 

56 

5 

555971 

5-46 

969909 

•81 

586o6a 

6-27 

4i3o38 

4i356i 

'4i3i85 

55 

6 

i 

556299 

5566a6 

5-45 

5-45 

969860 
96981 1 

• 81 
•81 

586439 
5868i 5 

6-27 

6-26 

54 

53 

556953 

5-44 

969762  : 

•81 

687190 

587566 

6-26 

412810 

52 

9 

557280 

5-44 

969714 

969065 

•81 

6-a5 

412434 

5i 

IO 

557606 

5-43 

•81 

687941 

6-a5 

412059 

5o 

»i 

5-43 

9-969616 

.82 

9-5883i6 

6-a5 

10  411684 

40 

12 

5-43 

969567 

.82 

588691 

589066 

6-24 

411309 

48 

i3 

558583 

5-42 

969518 

-82 

6-24 

4ioo34 

4] 

14 

558909 

5-42 

969469 

.82 

589440 

6-a3 

410560 

46 

i5 

559234 

5-41 

969420 

-8a 

589814 

6-23 

410186 

45 

16 

55o558 

5-4i 

969370 

•82 

590188 

6-a3 

409812 

44 

*7 

559883 

5-4o 

969321 

•82 

590562 

6-22 

409438 

43 

id 

560207 

5-4o 

969272 

-82 

590935 

691308 

6-22 

4ooo65 

42 

*9 

56o53i 

5-39 

969223 

• 82 

6-22 

408692 

4i 

20 

56o855 

5-39 

969173 

-8a 

691681 

6-21 

408319 

40 

ai 

9-561178 

5-38 

9-969124 

.82 

9 692064 

6-21 

10-407946 

407574 

3o 

aa 

56i5oi 

5-38 

969075 

.82 

592426 

6*20 

38 

23 

561824 

5-37 

969025 

.82 

592798 

6-20 

407202 

3? 

*4 

563146 

5.37 

968976 

.82 

593170 

6-19 

406829 

36 

25 

562468 

5-36 

968926 

• 83 

593542 

6- 19 

406458 

35 

26 

562790 

5-36 

968877 

• 83 

593914 

6- 18 

406086 

34 

a7 

563na 

5-36 

968827 

.83 

594285 

6- 18 

405715 

4o5344 

33 

2$ 

563433 

5-35 

968777 

968728 

968678 

.83 

594656 

6-i8 

32 

IS 

563755 

564075 

5-35 

5-34 

•83 

■83 

596027 

595398 

6- 17 
6-17 

404073 

404602 

3i 

3o 

3i 

9 • 564396 

5-34 

9-968628 

-83 

9-595768 

6-17 

io»4o423a 

29 

3a 

564716 

5-33 

968678 

-83 

596138 

6- 16 

4o386a 

28 

33 

565o36 

5.33 

968628 

-83 

596508 

6- 16 

403492 

27 

34 

565356 

5-3a 

968479 

-83 

596878 

6- 16 

403122 

26 

35 

565676 

5- 3a 

968429 

-83 

6*  1 5 

402753 

25 

36 

3-7  | 

565995 

5663i4 

5 - 3 1 

A 1. 

968379 

968329 

-83 
• 83 

6 - 1 5 

402384 

402016 

24 

23 

3*  Jl 

598354 

0*  ID 

38  I 
3, 

56663a 

5 - 3i 

968278 

-83 

6- 14 

401646 

22 

566961 

5-3o 

968228 

-84 

598722 

6-14 

401278 

21 

4o 

567269 

5-3o 

968178 

-84 

599091 

6- 13 

400909 

20 

4* 

9-567687 

5-29 

9-968128 

.84 

9 -599459 

6-i3 

io-4oo54i 

•9 

42 

567904 

568222 

! 5-29 

968078 

.84 

599827 

6- 13 

400173 

18 

43 

5-28 

968027 

•84 

600104 

6-15 

399806 

I? 

44 

568539 

5-28 

967977 

! .84 

6oo56a 

6-12 

399438 

l6 

45 

46 

568856 

569172 

569488 

! 5-28 

5-27 

967027 

967876 

.84 

.84 

600929 

601296 

6-11 

6-11 

390071  1 
398704  1 

1$ 

U j 

j 5-27 

967826 

-84 

601662 

6- 1 1 

398338 

! 3 j 

48 

49 

569804 

5-a6 

967775 

.84 

602029 

6- 10 

;2 

570120 

5-a6 

967725 

.84 

602396 

6-io 

1 1 

5o 

570435 

5-a5 

967674 

.84 

602761 

6-io 

1 397239 

10 

5i 

9-570751 

5-a5 

9-967624 

.84 

9-6o3i2T 

609 

10-396873 

9 

5a 

571066 

5-24 

967573 

.84 

603490 

6o3858 

6-09 

396507 

8 

53 

571380 

5-24 

967522 

.85 

6-09 

396142 

7 

54 

571696 

5-a3 

967471 

.85 

1 604223 

6-o8 

395777 

6 

55 

572000 
572324 
57 2636 

5-a3 

967421 

.85 

6o4588 

6-08 

3954ia 

5 

56 

h 

j 5-23 
5-22 

967370 

967319 

-85 
• 85 

6o4q53 

605317 

6 07 
6-07 

396047  1 
394683 

4 

3 

58 

672950 

! 5*22 

967268 

-85 

6o568a 

! 607 

6-o6 

394318 

2 

5-13263 

I 5-21 

96721”' 

-85 

606046 

39)054 

393690 

| 

60 

573575 

5-21 

967166 

-85 

606410 

6-o6 

0 

Cosine 

1 D. 

Sine 

1 D. 

Cotang. 

D. 

1 Tang. 

M. 

(68  DEORKRS.) 


{22  DECREES.)  A TABLE  OF  LOGARITHMIC 


40 


M. 

Sine 

D. 

Cowine 

D. 

Tang. 

D. 

Cotnng. 

60 

& 

S 

55 

54 

1 53 

5a 

5i 

5o 

4? 

fl 

45 

44 

43 

42 

4i 

40 

3* 

£ 

35 

34 

33 

3a 

3i 

3o 

£ 

2 

25 

24 

23 

22 

21 

20 

3 

3 

i5 
i4  i 
•3 

12 

11 

i® 

* 

5 

4 

3 

3 

0 

0 

1 

a 

3 

4 

5 

6 

I 

9 

10 

11 

ia 

13 

14 

15 

16 

3 

*9 

30 

31 

33 

33 

34 

35 

36 

3 

II 

3i 

3a 

33 

34 

35 

36 

ll 

39 

40 

41 
43 

43 

44 

45 

46 

U 

fo 

5i 

53 

53 

54 

55 

56 

ll 

& 

9*573575 

573888 

574300 

574513 

574834 

575i36 

575447 

575768 

576069 

576379 

576689 

9 576099 
577300 
577618 

5® 

578545 

578853 

579163 

579470 

579777 

9*58oo85 
58o39a 
580699 
58ioo5 
58i3 1 a 
58i6i8 
581934 
582339 
582535 
582840 

9*583i45 

583449 

583754 

584o58 

58436i 

584665 

584968 

585272 

585574 

585877 

9*586179 

58648a 

586783 

587085 

587386 

587688 

587989 

588289 

58859o 

58889o 

9 689190 
589489 
589789 
590088 
590387 
590680 
690984 

591282 
59i 58o 
691878 

5-21 

5*20 

5*20 

5*19 

5*19 

5*i8 

•>•'7 

1:3 

5*  16 

5*  16 

5 * 1 5 

5- 15 
5*14 

5*  14 

5*  i3 

5 • 1 3 

5*  i3 
5-12 

5*12 

5*  1 1 

5*  1 1 
5*u 
5*io 
5io 
5*09 
5.09 
5.09 
5*o8 

5-o8 

5*07 

5*07 

5.06 

5*o6 

5*  06 
5*o5 
5-o5 

5 04 

5- 04 

5*o3 

5*o3 

5*o3 

5-02 

5-02 

5-oi 

5*oi 

5-oi 

5*oo 

5*oo 

4-99 

4-99 

4-99 

4-98 

4-98 

4-97 

4-97 

4-97 

4-96 

496 

9967166 

967115 

967064 

967013 

966961 

966910 

966859 

966808 

966766 

966706 

966653 

9 966602 
96655o 
966499 
966447 
96639$ 
966344 
966292 
966240 
966188 
966136 

9*966085 

966o33 

966981 

965928 

965876 

965824 

965772 

965720 

965668 

9656i5 

9*965563 
9655i 1 
9654$8 
965406 
965353 
9653oi 
965248 
965195 
965 1 43 
965090 

9*965037 
964984 
96493 1 
964879 
964826 
964773 
964719 

964666 

964613 

964560 

9*964507 
964454 
964400 
964347 
064294 
964240 
964187 
964 1 33 
964080 
964036 

.85 

*85 

*85 

*85 

• 85 
*85 
85 
*85 
*86 
•86 
*86 

•86 

*86 

• 86 
*86 
*86 
*86 
*86 
*86 
*86 
*86 

•8? 

.87 

•87 

•*7 

•*7 

•87 

•87 

.87 

•87 

•87 

•«7 

? 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

*88 

.89 

• 89 
.89 

.89 

•89 

.89 

*89 

.89 

.89 

*89 

• 89 
.89 
.89 

9*606410 

606773 

607137 

607500 

607863 

1 608225 

6o8588 
608960 
609412 
609674 
6ioo36 

9610397 

6 1 07 D9 
611120 
61 1480 
61 1841 
612201 
6ia56i 
612921 
6i3a8i 
6i364i 

9*614000 
6i435o 
614718 
61 5077 

6 i 5435 

ass 

616509 

616867 

617224 

9*617582 

617939 

61829$ 

618653 

619008 

619364 

619721 

620076 

620442 

620787 

9*621142 

621497 

621852 

622207 

622561 

622915 

623269 

6a3623 

623076 

624440 

9 • 624683 
625o36 
6a5388 
625741 
636093 
636445 

626797 

627149 

627601 

627862 

6*  06 

6*  06 
6*o5 
6*o5 

6 04 
6*04 
604 
6*o3 
6*o3 

6 o3 
6*03 

6*02 

6*02 

6*oi 

6*oi 

6*oi 

6*oo 

6*oo 

6*oo 

5-99 

5.99 

5.98 

5.98 

5.98 

5.97 

5.97 

5.96 

5.96 

5*95 

5*96 

5*95 

5*94 

5*94 

5.94 

5*93 

5*93 

5*93 

5*92 

5*92 

5*92 

5*91 

5*91 

5*90 

5*90 

5*90 

5*89 

5*89 

5*89  i 

5*88 

5*88 

5*88 

5*87 

5*87 

5*87 

5*86 

5*86 

5*86 

5*85 

5*85 

10*393590 

393227 

| 39a863 

3925jo 
392137 
391775 
39141a 
391050 
390688 
3oo3a6 
389964 

10  389603 

389241 

38888o 

3885ao 

388169 

387709 

387449 

387079 

386719 

386359 

10 *386ooo 
385641 
385a8a 
3849a3 
384565 
384207 
383849 
388491 
383 1 33 
382776 

10*382418 
382061 
381705 
38 1 348 
380992 
380646 
380279 
379924 
379568 
379213 

10*378858 

3785o3 

378148 

377793 

377449 

37708$ 

376731 

376377 

376024 

375670 

1 0 - 375317 
374964 
374612 
374259 
373907 
373555 

373ao3 

372851 

372490 

372140 

Comij6 

D. 

Sine 

Dj^ 

Cotnng. 

I).  1 Tang.  J 

M. 

(G7  !)  K(1  M KKH  ) 


SINES  AND  TANGENTS.  (23  DEGREES.) 


41 


M. 

Sine 

D. 

Coaine 

D. 

Tang. 

D. 

Cotang. 

o 

9.591878 

4.96 

9 • 964026 

.89 

9-627852  1 

5-85 

10372148 

60 

i 

592176 

4.95 

963972 

• 89 

628203 

5-85 

37»797 

371446 

*9 

a 

592473 

4.95 

963919 

963865 

•89 

628554 

5-85 

58 

3 

592770 

593067 

4-95 

.90 

628906 

5-84 

371095 

57 

4 

4.94 

963811 

90 

629255 

5-84 

370745 

56 

5 

593363 

4.94 

963757 

90 

629606 

5-83 

370894 

55 

6 

59365o 

593955 

4.93 

963704 

.90 

^99“ 

63o3o6 

5-83 

370044 

64 

7 

4.93 

96365o 

.90 

5-83 

369694 

53 

8 

594251 

4.93 

963596 

.90 

63o656 

5-83 

369344 

5a 

9 

694547 

4-92 

963542 

.90 

63ioo5 

5-82 

368996 

5i 

10 

594842 

4.92 

963488 

.90 

63 1 355 

5-82 

368645 

5o 

li 

9 595i37 

4-91 

9-963434 

.90 

9-631704 

5-82 

10*368296 

49 

la 

595432 

4-91 

963379 

.90 

63ao53 

5-8i 

367947 

367699 

367260 

48 

i3 

595727 

4-91 

963325 

.90 

632401 

5-8i 

4J 

i*4 

596021 

4.90 

963271 

.90 

632750 

5-8i 

46 

i5 

5963i5 

4.90 

963217 

.90 

633098 

5- 80 

366902 

366553 

366205 

45 

16 

>7 

596600 

596906 

4.89 

4-89 

963 1 63 
963108 

.90 

.91 

633447 

633795 

5-8o 

5- 80 

44 

43 

18 

597196 

4-89 

963o54 

.91 

634143 

5-79 

365857 

42 

*9 

597490 

597783 

4-88 

962999 

962945 

.91 

634490 

634838 

5-79 

3655 10 

4i 

ao 

4-88 

91 

5-79 

365 1 62 

40 

21 

22 

9.598075 

598368 

4-87 

4-87 

9.962890 

962836 

•9i 

91 

9-635i85 

63553a 

5-78 

5-78 

io- 36481 5 
364468 

is 

23 

698660 

4*87 

962781 

91 

635879 

5-78 

364121 

37 

24 

698962 

4-86 

962727 

.91 

636226 

5-77 

363774 

36 

25 

699244 

4-86 

962672 

.91 

636572 

5*77 

363428 

35 

26 

599536 

4-85 

962617 

.91 

636919 

637265 

5-77 

363o8i 

34 

az 

599827 

600118 

4-85 

962562 

.91 

5-77 

362735 

33 

28 

4-85 

962508 

91 

637611 

5*76 

362389 

32 

29 

600409 

4-84 

962453 

.91 

637956 

638302 

5-76 

362044 

3i 

3o 

600700 

4-84 

962398 

.92 

5-76 

361698 

3o 

3i 

9 • 600990 
601 280 

4.84 

9.962343 

.92 

9-638647 

5-75 

io-36i353 

5? 

3a 

4-83 

962288 

.92 

638992 

5-75 

36ioo8 

33 

601570 

601860 

4-83 

962233 

.92 

639337 

5-75 

36o663 

27 

34 

4-82 

962178 

.92 

639682 

5-74 

36o3i8 

26 

35 

602 1 5o 

4-82 

962123 

.92 

640027 

5-74 

359973 

25 

36 

602439 

602728 

4-82 

962067 

.92 

640371 

5-74 

359629 

24 

h 

4*  81 

962012 

.92 

640716 

5-73 

359284 

23 

3$ 

39 

603017 

4-8i 

961957 

.92 

641060 

5-73 

358940 

22 

6o33o5 

4-8i 

961002 
96 1 846 

.92 

641404 

5-73 

358696 

21 

40 

603594 

4-8o 

92 

641747 

5*72 

358263 

20 

41 

42 

9. 6o3882 
604170 

4*  80 
4-79 

9-961701 

961735 

.92 

.92 

9-642091 

642434 

5-72 

5-72 

10-357909 

357566 

43 

604467 

4*79 

961680 

•93 

642777 

5-72 

357223 

1 *7 

44 

604745 

4-70 

4-7» 

961624 

•9j 

643120 

5-71 

35688o 

1 '0 

45 

6o5o32 

961560 

961613 

•93 

643463 

5-71 

356537 

10 

46 

6o53i9 

4-78 

•93 

6438o6 

5-71 

356194 

355852 

14 

4Z 

6o56o6 

i 4-78 

961468 

644148 

5-70 

i3 

48 

606892 

4-77 

961402 

•93 

644490 

644832 

5-70 

3555io 

12 

49 

606179 

4-77 

961346 

•93 

5.70 

355 1 68 

11 

5o 

606465 

4-70 

961290 

•93 

645174 

5-69 

354826 

10 

5: 

9*606751 

4-76 

q*96i235 

.93 

9-6455i6 

5-69 

10.354484 

8 

5a 

607066 

4-76 

961170 

• 93 

645857 

5-69 

354143 

53 

607322 

4-75 

961123 

*?3 

646199 

5.69 

3538oi 

7 

54 

607607 

4-75 

961067 

93 

646640 

5-68 

35346o 

6 

55 

607892 

608177 

4-74 

961011 

•93 

646881 

5-68 

353119 

352778 

352438 

5 

56 

4-74 

960055 

960890 

960843 

960786 

•93 

647222 

5-68 

4 

608461 

4-74 

•93 

647562 

5-67 

3 

56 

608745 

609029 

4-73 

4-73 

.94 

.94 

647903 

648243 

5-67 

5-67 

352097 

351767 

2 

1 

60 

6093 14 

4-73 

960730 

1 94 

648583 

5-66 

35i4i7 

0 

Cosine 

1 D. 

Sine 

1 D. 

Cotang. 

1 D. 

i Tang. 

M. 

(66  DEGREES.) 


(24  DEGREES.)  A TABLE  OF  LOGARITHMIC 


12 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Co  tang. 

0 

9609313 

4-73 

9-960730 

.94 

9-648583 

5-66 

10- 351417 

60 

i 

609597 

609880 

4-72 

960674 

•94 

648923 

; 5- 66 

351077 

5q 

s 

4-72 

960618 

.94 

649263 

5-66 

350737 

58 

3 

610164 

4-72 

96o56 1 

•94 

649602 

1 5-66 

350398 

J 57 

4 

610447 

4-7* 

96o5o5 

•94 

649942 

i 5-65 

1 35oo58 

1 ^ 1 

5 

610729 

4-71 

960448 

•94 

660281 

5-65 

3497*9 

349880 

1 55 

6 

611012 

4-70 

960392 

960335 

•94 

65o62o 

5-65 

1 ! 

7 

611294 

4-70 

•94 

650969 

5-64 

349041 

1 348703 

1 53 

8 

611576 

4.70 

960279 

•94 

65 1 297 
65 1 636 

5-64 

5a 

9 

61 i858 

4-6g 

960222 

•94 

5-64 

348364 

5i 

10 

612140 

4-69 

960165 

•94 

651974 

5-63 

348026 

5o 

ii 

9-612421 

4-6o 

0-960109 

• 95 

9-6523i2 

5-63 

10-347688 

49 

ia 

612702 

4-68 

960062 

-95 

65a65o 

5-63 

j 34735o 

48 

i3 

U 

612983 

6i3a64 

4-68 

4-67 

969995 

969938 

• 95 

652988 

6533a6 

5-63 

5-6a 

347012 

346674 

2 

i5 

6i3545 

4-67 

959882 

95 

653663 

5-6a 

346337 

45 

16 

6i38a5 

4-67 

4-66 

959826 

• 95 

664000 

5-62 

346000 

44 

*7 

6i4io5 

069768 

-95 

654337 

5 ■ 61 

345663 

43 

18 

614385 

4-66 

-95 

654674 

5-6i 

345326 

42  j 

*9 

6i4665 

4-66 

-95 

655oii 

5 - 61 

344989 

4* 

ao 

614944 

4-65 

959596 

-95 

655348 

5 - 61 

34465a 

4o 

ai 

9-6i5223 

4-65 

9-959539 

• 96 

9 • 655684 

5-6o 

io-3443i6 

3o 

aa 

6i55o2 

4-65 

959482 

-95 

656oao 

5- 60 

343o8o 

343644 

38 

a3 

615781 

4-64 

959425 

-95 

656356 

5- 60 

3]  1 

U 

616060 

4-64 

959368 

-95 

656692 

5-59 

3433o8 

36 

a5 

6i6338 

4-64 

969310 

.96 

657028 

5 • 59 

342972 

35  ! 

26 

616616 

4-63 

959253 

• 96 

657364 

5-59 

342036 

34 

27 

616894 

4-63 

959195 

.96 

657699 

5-59 

3423oi 

33 

28 

617172 

4-62 

969138 

.96 

658034 

5-58 

341966 

32 

29 

617460 

4-62 

969081 

.96 

658369 

5-58 

34163 1 

3i 

3o 

617727 

4-6a 

959023 

.96 

658704 

5-58 

341296 

3o 

3i 

9-618004 

4-6i 

4-6i 

9 • 958965 

.96 
• 96 

0 • 65qo3q 

5-58 

5-57 

10-340961 

20 

3a 

9 659373 

340627 

Ui 0201 

QjoQOO 

958850 

20 

33 

6i8558 

4-6i 

.96 

659708 

5.57 

340292 

27 

34 

6i8834 

4. 60 

968792 

958734 

958077 

• 96 

660042 

5-57 

339968 

20 

35 

619110 

4-6o 

.96 

660376 

5.57 

339624 

25 

36 

619386 

4- 60 

.96 

660710 

5-56 

339290 

338967 

3386a3 

24 

h 

619662 

4-59 

958619 

.96 

661043 

5-56 

23 

38 

619938 

4-59 

958561 

• 96 

661377 

5-56 

22 

39 

620213 

4-59 

9585o3 

•97 

661710 

5-55 

338290 

337967 

31 

40 

620488 

4-58 

958445 

•97 

662043 

5-55 

ao 

41 

9-620763 

4-58 

9-958387 

•97 

9-662376 

5-55 

10-337624 

*9 

42 

62io38 

4-57 

958329 

•97 

662709 

5-54 

337291 

336958 

18 

43 

6ai3i3 

4-57 

958271 

•97 

663o42 

5-54 

*7 

44 

621587 

4-57 

968213 

•97 

663376 

5-54 

3366a5 

16 

4< 

621861 

4-56 

958 1 54 

•97 

663707 

5-54 

336293 

335961 

i5 

4'- 

622135 

4-56 

958096 

968038 

•97 

664039 

5-53 

14 

42 

622409 

4-56 

•97 

664371 

5-53 

335629 

i3 

48 

622682 

4-55 

957979 

•97 

664703 

5-53 

335297 

334965 

334634 

12 

49 

50 

622956 

623229 

4-55 

4-55 

957021 

957863 

•97 

•97 

665o35  | 
665366 

5-53  j 
5-52 

11 

10 

5i 

9-6a35o2 

4-54 

9-967804 

•97 

9-665697 

5-52 

io-3343o3 

9 

5a 

623774 

4-54 

957746 

957687 

.98 

666029 

5-52 

333971 

8 

53 

624047 

4*54 

.98 

66636o 

5-5i 

333640 

7 

54 

624319 

4-53 

957628 

• 98 

666691 

5-5i 

333309  i 

6 

55 

62459* 

4-53 

957670 

.98 

667021 

5-5i 

332979 

5 

56 

624863 

4-53 

95751 1 

.98 

667352 

5-5i 

332648  ! 

4 

57 

6a5i35 

4-52 

967452 

• 98 

667682 

5-5o 

33a3i8 

3 

08 

59 

625406 

625677 

4-52 

4-52 

957393 

957335 

.98 

.98 

66801 3 
668343 

5-5o 

5 5o 

331987 

33i657 

i 

60 

625948 

4*5i 

957276 

.98 

668672 

5 5o 

33i3a8 

0 

1 

Coni  no 

I). 

Sino 

D. 

Co  tang. 

D.  | 

Tang. 

f65  DKOMEES.’l 


SINES  AND  TANGENTS.  '25  DEGREES.) 


M.  ! 

Sine 

D. 

Cosine 

D.  | 

Tang. 

D. 

Cotang,  j 

0 

9*626948 

4*5i 

9.957276 

•98 

9-668673 

5-5o 

io-33i327 

60 

I 

626219 

4-5i 

9572 1 7 

.98 

669002 

5*49 

330998  | 

50 

2 

626400 

4 - 5i 

957158 

.98 

669332 

5-49 

33o668  1 

58 

3 

626760 

4*5o 

957099 

.98 

669661 

5-4o 

33o339  I 

57 

4 

627030 

4*5o 

957040 

.98 

669991 

5.48 

330009  j 

56 

5 

627300 

4*5o 

956981 

•98 

670820 

5*48 

329680  | 

55 

6 

627670 

4*49  « 

936021 

•99 

670649 

5-48 

329351 

54 

7 

627840 

4*49  1 

966862 

•99 

670077 

5-48 

320023 

53 

8 

628109 

4*49  | 

9568o3 

•99 

671806 

5*47 

328694 

52 

9 

628378 

4*48 

956744 

•99 

671634 

5*47 

328366 

5i 

(0 

628647 

4.48 

966684 

•99 

671963 

5-47 

328037 

5o 

1 1 

9*628916 

4*47 

q. 95662 5 

•99 

>•672291 

5-47 

10-327709 

49 

n 

629185 

4*47 

956566 

•99 

672619 

5-46 

327381 

48 

(3 

629453 

4*47 

966606 

•99 

672947 

5*46 

327053 

47 

U 

629721 

4*46 

956447 

•99 

673274 

5-46 

326726 

46 

i5 

629989 

4.46 

956387 

•99 

673602 

5-46 

326398 

45 

16 

630257 

4.46 

966327 

•99 

673929 

5-45 

326071 

44 

• 7 

63o524 

4.46 

956268 

•99  ! 

674257 

5-45 

325743 

43 

18 

630792 

4*45 

966208 

1-00  j 

674584 

5-45 

3254i6 

42 

*9 

63 1 039 

4*45 

956148 

I -00  j 

674910 

5-44 

325090 

4i 

20 

63i326 

4*45 

956089 

1 -00  ; 

676237 

5-44 

3247W 

4o 

21 

9*63i5g3 

4*44 

9-966029 

1. 00 

9-675664 

5-44 

10-324436 

39 

22 

631839 

4*44 

955969 

I -00 

676890 

5-44 

324110 

38 

23 

632125 

4*44 

955909 

I -00  ! 

676216 

5-43 

323784 

37 

24 

632392 

4*43 

955849 

I -00 

676543 

5-43 

323457 

36 

25 

632658 

4.43 

955789 

I -00 

676869 

5-43 

323 1 3 1 

35 

26 

632923 

4*43 

955729 

I -00 

677194 

5-43 

322806 

34 

27 

633i89 

4*42 

955669 

I -oo 

67*-52o 

5-42 

322480 

33 

28 

633454 

4-42 

955609 

I *00 

67  1846 

5-42 

322154 

32 

29 

633719 

4*42 

955548 

I *00 

678171 

5-42 

321829 

3i 

3o 

633984 

4*4i 

955488 

I *00 

678496 

5-42 

32i5o4 

3o 

3i 

9*634249 

4.41 

9-955428 

I *01 

9*678821 

5 • 4 » 

10-321179 

32 

1 634514 

4.40 

955368 

I *01 

679146 

5-41 

320854 

28 

33 

1 634778 

4.40 

955307 

I -01 

679471 

5-41 

320520 

27 

34 

635042 

4.40 

955247 

1 01 

679795 

5*4i 

320205 

20 

35 

6353o6 

4*39 

955i86 

I -01 

680120 

5-4o 

319880 

25 

36 

635570 

4.39 

955126 

1 -01 

680444 

5*  40 

319556 

24 

37 

635834 

4'39 

955o65 

I -01 

680768 

5-4o 

310232 

23 

38 

636097 

4*38 

955oo5 

I -01 

681092 

5-4o 

318908 

22 

39 

63636o 

4*38 

964944 

I *01 

681416 

5.39 

318584 

21 

4o 

636623 

4*38 

I 954883 

1 -01 

681740 

5.39 

318260 

20 

4i 

9-636886 

4*37 

| 9 954823 

*•01 

9-682063 

5-39 

10*317937 

42 

637148 

4*37 

1 954762 

I -01 

682387 

5-39 

317613 

1§ 

43 

637411 

i 4*37 

I 964701 

I -01 

682710 

538 

317290 

»7 

44 

637673 

! 4 37 

954640 

I -01 

683o33 

5-38 

316967 

16 

45 

637935 

4*36 

i 964579 

I -01 

683356 

5-38 

? 16644 

i5 

46 

! 638197 

! 4-36 

9545 18 

I -02 

683679 

5-38 

3 i632 1 

; >4 

47 

i 638458 

4*36 

954457 

1 -02 

684001 

5-37 

3 1 5999 

| i3 

48 

j 638720 

4-35 

954396 

I • C2 

684324 

5-37 

3 1 56i6 

12 

49 

638981 

4*35 

964385 

I -02 

684646 

5-37 

3 1 5354 

1 1 

5o 

639242 

4*35 

954274 

I -02 

684968 

5.37 

3i5o32 

10 

5i 

9 *639603 

4*34 

1 9-954213 

1-02 

9-685290 

5-36 

io-3i47/o 

i 9 

52 

639764 

4*34 

954162 

I -02 

685612 

5-36 

3 14388 

! ® 

53 

640024 

, 4*34 

964090 

1-02 

685934 

5-36 

3 1 4066 

7 

54 

640284 

4*33 

954029 

I -02 

686255 

5-36 

3 i3745 

0 

55 

640544 

4*33 

953968 

I -02 

686577 

5-35 

3 1 3423 

5 

56 

640804 

4*33 

953906 

1 -02 

686898 

5-35 

3i3io2 

4 

*7 

641064 

4*32 

953845 

1 -02 

687219 

5-35 

312781 

3 

58 

64i324 

4*32 

' 953783 

1 -02 

687540 

5-35 

312460 

2 

59 

64i584 

4*32 

963722 

I -o3 

687861 

5-34 

3i2i3o 

1 

60 

641842 

I 4 • 3 1 

j 953660 

I -o3 

688182 

5 34 

3 i 1 818 

0 

Cosine 

1 D. 

Sine 

1 D. 

Cotang. 

D. 

. 

UJ 

(64  deorkkr.^ 


44 


(26  DEGREES.)  j-i.  TABLE  OF  LOGARITHM IO 


M. 

Sine 

D. 

Coftine 

D. 

Tang. 

D. 

Cotang. 

0 

9-641842 

4 • 3 1 

9-953660 

1 -o3 

9-688182 

5-34 

■o-3i 1818 

60 

i 

642101 

4 - 3 1 

953599 

& 

1 -o3 

6885o2 

5-34 

311498 

5o 

2 

642360 

4 - 3 1 

1 -o3 

688823 

5.34 

311177 

310807 

58 

3 

642618 

4-3o 

1 -o3 

689143 

5-33 

57 

4 

642877 

4-3o 

9534i3 

1 -o3 

689463 

5-33 

3 10537 

00 

5 

643 1 35 

4-3o 

953352 

1 -o3 

689783  1 

5-33 

110217 

55 

6 

643393 

4-3o 

963290 

1 -o3 

690103  | 

5-33  I 

309897 

54 

7 

64365o 

4-29 

953228 

1 -o3 

690423 

5-33 

309577 

309268 

^08938 

53 

ft 

643908 

4-29 

953 1 66 

1 -o3 

690742 

5-32 

> 62 

9 

644i65 

4-29 

963104 

1 -o3 

691062 

5-32 

5i 

IC 

644423 

4-28 

953042 

1 -o3 

691 38 1 

5-32 

308619 

5o 

9 • 644680 

4 28 

9-962980 

1 -04 

9-691700 

5 - 3 1 

io-3o83oo 

4o 

12 

644936 

4-28 

952918 

952855 

1 -04 

692019 

5 • 3 1 

30798 1 

48 

i3 

645 io3 

4-27 

1 -04 

692338 

5 - 3 1 

307662 

47 

14 

646460 

4-27 

952793 

962781 

M -o4 

692656 

5 • 3 1 

307344 

46 

i5 

645706 

4-27 

1 -04 

692975 

5 • 3 1 

307025 

45 

16 

646962 

4-26 

952669 

1 -04 

693293 

5-3o 

306707 

3o6388 

44 

*7 

6462 1 8 

4-26 

962606 

1 -04 

693612 

5*3o 

43 

ift 

646474 

4-26 

952544 

1 -04 

693930 

5-3o 

3o6oio 

42 

•9 

646729 

4-25 

952481 

1 -04 

694248 

5-3o 

306702 

4i 

20 

646984 

4-25 

952419 

.•04 

694566 

5*29 

3o5434 

40 

21 

9-647240 

4-25 

9952356 

1 -04 

9 • 694883 

5*29 

io-3o5i 17 

32 

22 

647494 

4-24 

952294 

952231 

1 -04 

695201 

5 -.29 

304799 

38 

23 

647749 

648004 

4-24 

1 -04 

6955 1 8 

5-29 

304482 

37 

24 

4-24 

962168 

1 -o5 

695836 

5-20 

3o4i64 

36 

25 

648258 

4-24 

962106 

1 -o5 

696153 

5-28 

3o3847 

35 

26 

6485 1 2 

4-23 

952043 

1 -o5 

696470 

5-28 

3o353o 

34 

27 

648766 

4-23 

951980 

1 -o5 

696787 
697 I 03 

5-28 

3o32i3 

33 

2ft 

649020 

4-23 

1 -o5 

5-28 

302897 

32 

649274 

4-22 

1 -o5 

697420 

5-27 

3o258o 

3i 

3o 

649527 

4-22 

951791 

1 -o5 

697736 

5-27 

302264 

3o 

3i 

9-649781 

4-22 

9-951728 

1 -o5 

9-698063 

5-27 

10.301947 

3oi63i 

29 

32 

65oo34 

4-22 

961666 

1 -o5 

698369 

698685 

5-27 

28 

33 

060287 

4-21 

951602 

1 -o5 

5-26 

3oi3i5 

27 

34 

65o539 

4-21 

961639 

1 -o5 

69900 I 

5-26 

300999 

26 

25 

35 

650792 

4-21 

951476 

1 -o5 

6993 1 6 

5-26 

300684 

36 

65 1 044 

4-20 

961412 

1 -o5 

699632 

5-26 

3oo368 

24 

37 

65 1 297 

4-20 

95i349 

1 -06 

699947 

5-26 

3ooo53 

23 

3ft 

65 1 549 

4-20 

951286 

1 -06 

700263 

5-25 

299737 

22 

39 

65 1 800 

4- 19 

951222 

1 -06 

700578 

5-25 

299422 

21 

4o 

652052 

4- 19 

961159 

1 -06 

700893 

5-25 

299107 

20 

4i 

9-6523o4 

4-19 

9951096 

95ioJ2 

1 -06 

9-701208 

5-24 

io- 298792 

>9 

42 

652555 

4-ift 

1 -06 

7oi523 

5-24 

18 

43 

652806 

4- 18 

950968 

1 -06 

701837 

5-24 

*7 

44 

653o57 

4- 18 

95ooo5 

950841 

1 -06 

702152 

5-24 

297848 

16 

45 

6533o8 

4*i8 

1 -06 

702466 

5-24 

297534 

i5 

46 

653558 

4-17 

950778 

1 -06 

70278c 

5 23 

297220 

14 

47 

6538o8 

4-17 

950714 

1 -06 

703095  1 

5-23 

296900 

296091 

i3 

4ft 

654059 

4-n 

95o65o 

1 -06 

703409  1 

5-23 

12 

49 

654309 

654558 

4- 16 

95o586 

1 -06 

*93720 

5-23 

296277 

11 

5o 

4- 16 

95o522 

1-07 

704036 

5-22 

295904 

10 

5i 

9-654808 

4- 16 

9-95o458 

107 

9-704350 

5-22 

1 )• 296650 

9 

52 

53 

655o58 

655307 

4- 16 

4 - 1 5 

95o3o4 

95o33o 

107 

1-07 

704663 

704977 

5-22 

5-22 

295337 

295023 

8 

7 

54 

655556 

4 - 1 5 

950266 

1.07 

706290 

5-22 

294710 

294807 

294084 

6 

55 

6558o5 

4 * 1 5 

960202 

1 07 

7o56o3 

5-21 

5 

56 

656o54 

4-i4 

95oi 38 

1-07 

705916 

5-21 

4 

*7 

656302 

4-14 

950074 

1-07 

706228 

5-21 

293772 

3 

5ft 

65655i 

4-i4 

950010 

1-07 

706541 

5-21 

293460 

2 

59 

656799 

4-i3 

949945 

949881 

1 ‘ 07 

706864 

5-21 

293140 

1 

6o 

657047 

4 - 1 3 

1-07 

707166 

5-20 

292834 

0 

CoHiiie 

D. 

Sine 

D. 

Cotnng. 

D.  1 

1 Tang. 

(63  DKOKKKB.8 


SINES  AND  TANGENTS.  (27  DEGREES.) 


46 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotnng. 

0 

9.657047 

657395 

4*  i3 

9.949881 

1-07 

9.707166 

5-20 

1 10- 292834 

60 

i 

4- 13 

949816 

1-07 

707478 

5-20 

, 392522 

J9 

a 

65754a 

4*12 

949752 

1.07 

7°779o 

708102 

708414 

5-20 

1 292210 

58 

3 

4 

3$ 

4*12 

4- 12 

949688 

949623 

1 -o8 

1 -o8 

5-20 

5.19 

291898 
! 291606 

3 

5 

658384 

4*12 

949558 

1 -08 

708726 

5-19 

| 291274 

50 

6 

65853i 

4ii 

949494 

1 *o8 

709037 

5*19 

290963 
29065 1 

54 

7 

658778 

4- 11 

949429 

1-08 

709349 

! 5*19 

53  ! 

6 

659035 

4*ii 

949364 

1 *08 

709660 

5- 19 

290340 

5a 

9 

10 

659371 

659517 

4*io 

4*io 

949300 

949235 

1 -08 
1-08 

70997* 

710282 

5- 18 
5.i8 

I 390029 
289718 

5i 

5o 

u 

9-659763 

4*  10 

9.949170 

1 *o8 

9.710593 

5-i8 

10-289407 

289096 

49 

ia 

660009 

66oa55 

4-09 

949105 

i-o8 

710904 

5 • 18 

48 

i3 

4*09 

949040 

1 -o8 

711215 

5- 18 

288785 

47 

14 

66o5oi 

4-09 

948975 

1 -o8 

71 i5a5 

5- *7 

288475 

46 

i5 

660746 

4*09 

948910 

948845 

1 -08 

711836 

5.17 

288164 

45 

16 

660991 
661 a36 

4*  08 

1 *o8 

712146 

5.17 

287854 

44 

*7 

4- 08 

948780 

1 -09 

712456 

5-17 

287544 

43 

id 

661481 

4*o8 

948715 

1 *09 

712766 

5.16 

287234 

43 

*9 

661736 

4-07 

94865o 

1 >09 

713076 

5- 16 

286924 

286614 

4i 

30 

661970 

4-07 

948684 

1 -09 

7i33% 

5.i6 

40 

31 

9*663214 

4*07 

9.948519 

1-09 

9 - 7 1 3696 

5-  *6 

io-2863o4 

39 

as 

a3 

662459 

662703 

4*07 

4-o6 

948454 

948388 

1-09 

1-09 

714006 

7*43i4 

5.16 

5 • 1 5 

285996 

285686 

38 

37 

24 

662946 

4- 06 

948323 

1 -09 

714624 

5**5 

285376 

36 

35 

663190 

4- 06 

948257 

1*09 

714933 

5 • 1 5 

285067 

35 

36 

663433 

4-o5 

948192 

1 09 

716242 

5 • 1 5 

284758 

34 

27 

663677 

4*o5 

948126 

1 -09 

7 1 555i 

5i4 

284449 

33 

a 8 

663920 

4*o5 

948060 

1-09 

7i586o 

5-14 

284140 

32 

29 

664163 

4-o5 

947995 

1 • 10 

716168 

5-i4 

283832 

3i 

3o 

664406 

4*o4 

947929 

I-IO 

716477 

5-14 

283523 

3o 

3i 

9-664648 

4-o4 

9-947863 

1 • 10 

9.716785 

5- *4 

IO-2832I5 

29 

3a 

33 

664891 
665 1 33 

4*o4 

4-o3 

947797 

9477^1 

1*10 

I-IO 

7*7093 

7*74oi 

5i3 

5 • 1 3 

282907 

282099 

28 

34 

"'*665375 

4-o3 

047665 

I . 10 

717709 

718017 

5 - 1 3 

282291 

26 

35 

665617 

4*o3 

947600 

I . 10 

5- *3 

281983 

25 

36 

665859 

4-oa 

947533 

I • 10 

7i83a5 

5 • 1 3 

281670 

34 

37 

666100 

4*02 

947467 

I . 10 

718633 

5.12 

281367 

23 

3 8 

666342 

4*02 

947401 

I-IO 

7 1 8940 

5-12 

281060 

22 

39 

666583 

4-02 

947335 

I-IO 

719248 

5.12 

280752 

21 

4o 

666824 

4-oi 

947269  ' 

I . 10 

719555 

5-12 

280445 

30 

4i 

9-667065 

4-oi 

9-947203 

1 . 10 

9.719862 

5-12 

io-28oi38 

4a 

667305 

4-oi 

947136 

I "II 

720169 

720476 

720783 

721089 

5.  II 

279831 

|8 

43 

667546 

4-oi 

947070 

III 

5- 1 1 

279524 

*7 

44 

45 

667786 

668027 

4-oo 

4-oo 

947004 

946937 

I'M 

I'll 

5- 11 
. 11 

XI 

16 

10 

46 

668267 

4-oo 

94687 1 

III 

731396 

5- 11 

278604  i 

14 

47 

6685o6 

3 ’99 

946804 

1 . 1 1 

721702 

5io 

278298 

i3 

4b 

49 

668746 

668986 

3‘99 

3-99 

946-738 

946071 

I -11 

1 . 1 1 

722009 

7223i5 

5- 10 
5- 10 

IX 

12 

11 

5o 

669225 

3*99 

946604 

1 . 1 1 

722621 

5-10 

277379 

. *0 

5i 

9-669464 

3.98 

9 • 946538 

1 . 1 1 

9.722937 

5*  10 

,0'$$ 

9 

5a 

669703 

3.98 

94647* 

I'll 

723232 

509 

8 

53 

669942 

3.98 

946404 

I • 1 1 

723538 

5.09 

376462 

7 

54 

670181 

3 '91 

946337 

1 . 1 1 

723844 

509 

276156 

6 

55 

670419 

67o658 

i'91 

946270 

1 - ia 

724149 

5.09 

27585i 

5 

56 

3'97 

946203 

1*12 

724454 

5*09 

5.08 

275546 

4 

670896 

3-97 

946136 

1*12 

375241 

3 

5$ 

671134 

3.96 

946069 

112 

5- 08 

274935 

374631 

a 

59 

671372 

3.96 

946002 

1-12 

725369 

5-o8 

1 

6o 

671609 

3.96 

945935 

I • 1 2 

725674 

5o8 

274326 

0 

1 

Cosine 

D. 

Sine 

D. 

Cotang. 

D. 

1 Tang. 

M. 

(62  DEGwercs.) 


(2H  DEGREES.)  A TABLE  OF  LOGARITHMIC 


Hi 


M. 

Sine 

D. 

Cortine 

D. 

Tang. 

D. 

Cotang. 

0 

9671609 

3-96 

9-945o35 

946868 

1 - 1 a 

9-725674 

5 08 

110-274326 

60 

i 

671847 

3-95 

1 • 12 

725979 

726284 

5 08 

274021 

59 

2 

672084 

3 -95 

946800 

1 • 12 

1 5‘°7 

273716 

1 58 

3 

672321 

3-95 

945733 

945o66 

1 • 12 

,26588 

726892 

' 5*07 

1 273412 

' 57 

4 

672668 

3-95 

I -12 

5-07 

' 273108 

56 

5 

672795 

3-94 

9455o8 

945531 

I - 12 

I 727'97 

5-0-) 

| 272803 

55  ! 

6 

6730J2 

3-94 

I • 12 

i 727501 

1 5'°7 

272499 

54  i 

7 

673268 

3-94 

945464 

i • 1 3 

727805 

1 5-o6 

272195 

53  ; 

8 

6735o5 

3-94 

945396 

1 • i3 

728109 

5- 06 

271801 

271588 

5a 

9 

673741 

3-93 

945328 

1 • i3 

728412 

1 5 06 

5i 

lO 

6739-- 

3.93 

945261 

1 • i3 

728716 

5-o6 

27 1 284 

5o 

1 1 

9 674213 

3.9? 

9-945193 

1 • i3 

9-729020 

5-o6 

10-270980 

40 

ia 

674448 

3.92 

945 1 25 

1 • i3 

729323 

5-o5 

270677 

48 

1 3 

674684 

3-92 

945o58 

1 • i3 

i 729626 

5-o5 

270374 

47 

14 

674919 

3-92 

944990 

1 • i3 

729929 

73oa3j 

5-o5 

270071 

46 

i5 

675i55 

3-92 

944922 

944854 

1 - 1 3 

5-o5 

269767 

45 

16 

675390 

3-91 

1 • 1 3 

73o535 

5-o5 

269465 

44 

'7 

675624 

3-91 

944786 

1 • i3 

73o838 

5-o4 

269167“ 

*"43 

i8 

675859 

3-91 

9447*8 

1 • i3 

73ii4i 

5-o4 

26885g 

42 

>9 

676094 

3-91 

94465c 

1 • i3 

731444 

5-o4 

268556 

41 

ao 

676328 

3-90 

944582 

i- 14 

731746 

5-04 

268254 

40 

21 

9-676562 

3-90 

9-9445i4 

1 • 14 

9-732048 

5-o4 

10-267952 

39 

a2 

676796 

3-90 

944446 

1 • 14 

73a35 1 

5-o3 

267649 

38 

a3 

677040 

3-90 

944377 

1 • 14 

732653 

5-o3 

267347 

267046 

266743 

37 

24 

677264 

3.89 

944309 

1 • 14 

732955 

5-o3 

36 

a5 

677498 

3-89 

944241 

1 • 14 

733257 

1 5-o3 

35 

26 

67773 1 

3-89 

944172 

1 • 14 

733558 

5-o3 

266442 

34 

27 

677964 

3-88 

944104 

114 

733860 

5-02 

266140 

33 

28 

6784^0 

3-88 

944o36 

1 14 

734162 

5-02 

265838 

3a 

29 

3-88 

943967 

943899 

1 • 14 

734463 

5-02 

265537 

3i 

3o 

678663 

3-88 

1 • 14 

734764 

5-02 

265236 

3o  | 

3i 

9-678896 

3-87 

9 • 943830 

1 • 14 

9- 735o66 

5-02 

10-264934 

29 

3a 

679128 

3-87 

94376i 

1 • 14 

735367 

5-oa 

264633 

28 

33 

679360 

3-87 

943693 

1 • i5 

735668 

5-oi 

264332 

27 

34 

679692 

3-87 

943624 

1 • 1 5 

735969 

5*oi 

26403 1 

26 

35 

679S24 

68oo56 

3-86 

943555 

1 • i5 

736269 

5-oi 

263731 

25 

36 

3-86 

943486 

1 * 1 5 

736570 

5*oi 

26343o 

24 

37 

680288 

3-86 

943417 

943348 

1 - 1 5 

736871 

5-oi 

263129 

23 

38 

680519 

3-85 

i * 1 5 

737171 

5-oo 

262829 

22 

39 

680750 

3-85 

943279 

1 • 1 5 

737471 

5-oo 

262529 

21 

40  1 

680982 

3-85 

943210 

1 - 1 5 

737771 

5-oo 

262229 

20 

4 1 

9681213 

3-85 

9943i4i 

1 - 1 5 

9-738071 

5-oo 

10-261929 

*2 

4a 

68i443 

3-84 

943072 

1 - 1 5 

738371 

5-oo 

261629 

l8 

43 

681674 

3-84 

943oo3 

1 • i5  , 

738671 

4-99 

261329 

'7 

44 

68mo5 

3-84 

942934 

942864 

1 - 1 5 I 

738971 

4-99 

261029  | 

|6 

i5 

<5 

682135 

3-84 

i • i5 

739271 

4-99 

260729  | 

i 46 

682365 

3-83 

94279s 

1 • 16 

739670 

4-99 

260430  ! 

14 

47 

682595 

3-83 

942726 

1 • 16 

739870 

4-99 

260 i 3o  1 

3 1 

48 

682825 

3-83 

942656 

1 16 

74016c 

4*99 

259831  1 

12 

49 

683o55 

3-83 

942587 

1 • 16 

74046c 

4-98 

25^532  | 

1 1 

5o 

683284 

3-82 

942517 

1 • 16 

740767 

4-98 

269233  i 

10 

1 

5i 

9 6835i4 

3-82 

9-942448 

1 • 16. 

9-741066 

4-98 

io- 258934 

S 

5a 

683743  l 

3-82 

942378 

1 • 16 

741365 

4-98 

258635 

8 

53 

683972  i 

3-82 

9423o8 

1 ■ 16 

741664 

4-98 

258336 

7 

54 

684201 

3-8i 

942239 

1 • 16 

741962 

4*97 

258o38 

6 

55 

684430  1 

3 • 81 

942169 

1 • 16 

742261 

4-97 

257739 

5 

56 

684658  i 

3 • 8 1 

942099 

1 • 16 

742559 

742858 

4-97 

257441 

4 

*7 

684887  ' 

3- 80 

942029 

1 • 16 

4-97 

257142 

256844 

3 

58 

685ii5  1 

3- 80 

941959 
94 1 889 

1 • 16 

743 1 56 

4-97 

2 

59 

685343  | 

3- 80 

117 

743454 

4-97 

4-90 

256546 

1 

60 

68557 1 1 

3- 80 

941819 

- 17 

743752 

256248 

0 

1 

(Wmc  1 

I). 

Sine 

D. 

Cotang. 

d.  i 

__Tang. 

M. 

(61  DRORRRB.) 


SINES  AND  TANGENTS.  (29  DEGREES.; 


47 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

| Cotang. 

r 

i 

0 

9 68557 1 

3- 80 

9941819 

II7 

9^43752 

4.96 

1 10 -256248 

! 60 

i 

685799 

3-79 

941749 

1*17  ! 

744o5o 

4.96 

25595o 

2 

686027 

3-79 

941679 

1 * 17 

744348 

4.96 

255652 

1 1 

3 

686264 

3-79 

941609 

1 * 17 

744645 

4-96 

255355 

5] 

4 

686482 

3-79 

94i539 

i-n 

744943 

4-96 

255o57 

56  ! 
55  I 

5 

686709 

686936 

3.78 

941469 

1 • 17 

745240 

4-96 

254760 

6 

3-78 

941398 

117 

745538 

4-96 

254462 

1 54 

7 

687i63 

3-78 

941328 

* * 17 

745835 

4-9^ 

254i65 

53 

8 

687)89 

687616 

3-78 

941258 

1-17 

746132 

j 4-95 

253868 

52 

9 

3-77 

941187 

1 * 17 

746429 

4.95 

253571 

5i 

IO 

687843 

3 * 77 

941117 

117 

746726 

4.95 

253274 

5o 

II 

9 • 688069 
688296 

3.77 

9-941046 

1 • 18 

9-747023 

4-94 

10  252977 

42 

12 

3-77 

940975 

1 - *8 

747319 

4-94 

252o8i 

! 48 

i3 

688521 

3-76 

940905 

940834 

1 - 18 

747616 

4-94 

252384 

! 47 

14 

688747 

3-76 

1 • 18 

7479*3 

748209 

4.94 

252087 

46 

i5 

688972 

3.76 

940763 

940693 

1 • 18 

4-94 

251791 

i 45 

16 

689198 

3.76 

1*18 

7485o5 

4.93 

25i495 

! 44 

*7 

689423 

3-75 

940622 

118 

748801 

4.93 

20 1 I9Q 

43 

18 

689648 

3.75 

94o55 1 

1 • 18 

749097 

4.93 

2-50903 

42 

*4 

689873 

3.75 

940480 

1 • 18 

7493o3 

749689 

4.93 

1 25o£p7 

4i 

20 

690098 

3*76 

940409 

1 • 18 

4.93 

25o3i 1 

! 40 

21 

9' 690323 

3-74 

9>94o338 

1 • 18 

9-749985 

4.93 

io-25ooi5 

*9 

22 

690548 

3-74 

940267 

1 • 18 

760281 

4-92 

2497 '9 

38 

23 

690772 

3-74 

940196 

1 • 18 

750576 

4-92 

249424 

37 

24 

690996 

3-74 

940125 

119 

750872 

4-92 

249128 

36 

25 

691220 

3-73 

940054 

119 

751167 

4-02 

248833 

35 

26 

691444 

3.73 

939982 

1 19 

751462 

4-92 

248538 

34 

691668 

3-73 

939911 

119 

751757 

4-92 

248243 

33 

28 

691892 

3-73 

939840 

1 ’ 19 

752o52 

4.91 

247948 

32 

»9 

692115 

3-72 

939768 

119 

752347 

4-91 

247653 

3i 

3o 

692339 

3-72 

939697 

i*i9 

752642 

4-91 

247358 

3o 

3i 

9 692562 

3.72 

9-939625 

1 ' 19 

9-762937 

4.91 

io-247o63 

29 

32 

692785 

3-71 

939554 

119 

75323 1 

4-91 

246769 

28 

33 

693008 

3.71 

939482 

119 

753526 

4.91 

246474 

27 

34 

693231 

3-71 

939410 

1-19 

753820 

4.90 

246180 

26 

35 

693453 

3.71 

939339 

1 * 19 

754ii5 

4-9° 

245885 

25 

36 

693676 

3-70 

939267 

I -20 

754400 

754703 

4.90 

245591 

24 

oZ 

693898 

3-70 

939195 

1-20 

4.90 

245297 

23 

38 

694120 

3-70 

939123 

1-20 

754997 

4.90 

245oo3 

22 

39 

40 

694342 

694564 

3-70 

3-69 

939052 

938980 

1-20 

1-20 

755291 

755585 

4-9° 

4.89 

244709 

244410 

21 

20 

4i 

9 -694786 

3-69 

9-938908 

1-20 

9-755878 

4.89 

10-244122  | 

IQ 

42 

696007 

3-69 

938836 

I -20 

756172 

4.89 

243828  , 

l8 

43 

695229 

3-69 

938763 

938691 

1-20 

756465 

4.89 

243535  1 

•7 

44 

695450 

3-68 

I -20 

756759 

4.89 

243241  I 

16 

45 

695671 

3-68 

938619 

1-2C 

757052 

4-89 

242948  ' 

i5 

46 

j 696892 

3-68 

938547 

938470 

1-20 

757345 

4.88 

242655  ■ 

14 

47 

1 696113 

3-68 

I -20 

757638 

4.88 

242362 

i3 

48 

! 696334 

3-67 

938402 

I -21 

757931 

758224 

4 • 88 

242069 

12 

49 

; 690554 

3-67 

93833o 

I -21 

4-88 

241776 

11 

5o 

696776 

3-67 

938258 

I -21 

7585i 7 

4-88 

241483 

to 

5i 

9-696996 

3-o7 

9-938i85 

ill 

9-758810 

4.88 

10-241 190 

9 

5a 

697215 

3-66 

9381 i3 

I -21 

759102 

4.87 

240898 

8 

53 

697435 

3-66 

938040 

1-21 

759395 

759687 

4.87 

240605 

7 ‘ 

54 

697664 

3-66 

937967 

93789$ 

I -21 

4.87 

2403 1 3 

6 1 

55 

697874 

698094 

3-66 

I -21 

769979 

4.87 

240021 

5 

56 

3-65 

937822 

I -21 

760272 

760564 

4-87 

239728 

4 

*z 

698313 

3-65 

937749 

937676 

I -21 

4-8  7 

239436 

3 

58 

698532 

3-65 

I -21 

760856 

4-86 

23oi44 

2 

59 

698751 

3-65 

937604 

•21 

761 148 

4-86 

23885a 

1 

60 

698970 

3-64 

937531 

I -21 

761439 

4-86 

a3856i 

0 

Cosine 

D. 

1 Sine 

D. 

Cotang. 

1 D. 

I Tang. 

M. 

27 


(60  DEGRESS.) 


48 


(SO  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 

Sino 

D. 

Cosine  | D. 

Tang. 

D. 

Cotang. 

0 

1 

3 

3 

4 

5 

1 6 
i 

9 

10 

13 

13 

14 

15 

16 

\l 

*9 

20 

21 

33 

23 

24 

25 

26 

3 

5 

31 

32 

33 

34 

35 

36 

ll 

3, 

40 

41 

42 

43 

44 

45 

46 

% 

5o 

5c 

52 

53 

54 

55 

56 

U 

6 

9-698970 

699189 

699407 

699620 

699844 

100062 
700280 
700498 
/00716 
700933 
701 1 5i 

9701368 

701585 

701802 

702019 

702236 

702452 

702669 

702885 

703101 

703317 

9-703533 

703749 

703964 

704179 

704395 

704610 

704826 

706040 

7o5254 

705469 

9 • 705683 
705898 
706112 
706326 
706539 
706753 
706967  1 
707180  , 
707393 
707606 

9-707819 

708032 

708245 

708458 

708670 

708882 

709094 

709306 

709518 

709730 

9-709941 

7ioi53 

7 1 o364 
710575 
710786 
710997 
711208 
7»i4i9 
711629 
711839 

3-64 

3-64 

3-64 

3-64 

3-63 

3-63 

3-63 

3-63 

3-63 

3-62 

3-62 

3-62 

3-62 

3- 61 
3-6i 
3-6i 
3-6i 

3- 60 

3- 60 
3-6o 
3-6o 

3-59 

3-59 

3-59 

3-59 

3 - 5o 
3-58 
3-58 
3-58 
3-58 
3-57 

3-57 
3-5  7 
3-57 
3-56 
3-56 
3-56 
3-56 
3-55 
3-55 
3-55 

3-55 

3-54 

3-54 

3-54 

3-54 

3-53 

3-53 

3-53 

3-53 

3-53 

3-52 

3*52 

3-52 

3-52 

3 - 5i 
3-5i 
3*5i 

3 - 5 1 
3-5o 
3-5o 

9 • 93753 1 

937458 

937385 

937312 

937238 

937165 

937092 

937019 

936946 

936872 

936799 

9-936725 
936652 
936578 
9365o5 
93643 1 
936357 
936284 
936210 
936 1 36 
936062 

9-935988 

935914 

935840 

935766 

935692 

935618 

935543 

935469 

935396 

935320 

9-935246 

935171 

935097 

g35o22 

934948 

934873 

934798 

934723 

934649 

934574 

9-934499 

934424 

934349 

934274 

934199 

934123 

934048 

933o73 

933898 

933822 

9-933747 

933671 

933596 

933520 

933445 

93336o 

933293 

933217 

933i4i 

933o66 

1 1-21 
1-22 
1-22 

1-22 
i 1-22 
1-22 
1-22 
1-22 

1-22 

1-22 

1-22 

1-22 

1-23 

I - 23 

I - 23 
! 1 - 23 

1-23 

1 - 23 
1-23 
1-23 
1-23 

1-23 

1-23 

1-23 

1-24 

1-24 

1-24 

1-24 

1-24 

1-24 

1-24 

1-24 

I • 24 

1-24 

1-24 

1-24 

1-24 

I -25 

1 - 25 

1 -25 

1 -25 

1-25 

I -25 
1-25 

I - 25 
1-25 

I -25 

I -25 

I -25 

I -26 

I • 26 

I -26 
1-26 
1-26 
1-26 

I • 26 
1-26 

I -26 

I -26 
1-26 
1-26 

9-761439 

761731 

762023 

1 762314 

762606 
762897 
763 I 88 
763419 
763770 
764061 
764352 

9 • 764643 
764933 
765224 
765514 
7658o5 
766095 
766385 
766675 
766965 
767255 

9-767545 

767834 

768124 

768413 

768703 

768902 

769281 

769570 

769860 

770148 

9.770437 

770726 

771015 

77i3o3 

771502 

771880 

772168 

772457 

vVii 

773o33 

9-773321 

773608 

773806 

774i84 

774471 

774709 

775046 

775333 

775621 

775908 

9-776iq5 

776482 

776769 

777055 

77734a 

777628 
77791 5 
778201 
778487 
778774  I 

4-86 

4-86 

4-86 

4-86 

4-85 

4-85 

4-85 

4-85 

4-85 

4-85 

4-84 

4-84 

4-84 

4-84 

4-84 

4-84 

4-84 

4-83 

4-83 

4-83 

4-83 

4-83 

4-83 

4-82 

4-82 

4-82 

4-82 

4-82 

4-82 

4-8i 

4-8i 

4-8i 

4-8i 

4 • 81 

4-81 

4-8i 

4-8o 

4- 80 

4 -8o 

4- 80 

4- 80 

4- 80 

4-79 
4-79 
4-79 
4-79  j 
4-79  ' 
4-79  ' 

4-79  i 

4-78 

4-78 

4-78 

4-78 

4-78 

4-78 

4-78 

4-77 

4-77 

4-77 

4-77 

4-77 

io-23856i 

238269 

337Q77 

237686 

337394 

237103 

236812 

236521 

23623o 

235939 

235648 

10. *35357 
235067 
234776 
234486 
234195 
233905 
2336i5 
233325 
233o35 
232745 

10-232455 
232166 
231876 
23 1 687 
231297 
23 1 008 

230719 

23o43o 

23oi4o 

229852 

10-229563 

229274 

228986 

228697 

228408 

228120 

227832 

227643 

227255 

226967 

10-226679 

226392 

226104 

2258i6  1 
225529  j 
225241  1 
224964  | 
224667 
224379 
224092 

io-2238o5 
2235i8 
22323 1 1 
222045  i 
222658 
222372 
222085 
221799 

2 2 1 5 1 2 
221226 

60 

3 

55  I 
54  ! 
53  ; 
52 

5i 

5o 

3 

3 

45 

44 

43 

43 

41 

40 

3 

£ 

35 

34 

33 

32 

3i 

3o 

3 

3 

25 

24 

23 

22 

21 

20 

:? 

3 

i5 

14 

iJ 

:;! 

10 

g 

5 

4 

3 

2 

1 

0 

Cosine  1 

D. 

Sino 

D. 

OotaiiR.  j D. 

TtU1g- 

M. 

(50  DEGREES.) 


SINES  AND  TANGENTS.  (31  DEGREES.) 


49 


M. 

Sine 

D. 

Cosine 

D. 

Tang,  j 

D. 

Cotang. 

0 

9-711839 

3-5o 

9-933066 

1-26 

9-778774 

4-77 

10-221226 

60 

i 

7i3o5o 

3-5o 

932990 

1-27 

779060 

4*77 

220940 

*9 

2 

712260 

3-5o 

9329i4 

1-27 

779346 

4-76 

220654 

58 

3 

712469 

3-49 

932838 

1-27 

779632 

4-76 

22o368 

57 

4 

712619 

3-49 

932762 

1.27 

779918 

4-76 

220082 

56 

5 

7 1 2880 

3-49 

932685 

1-27 

780203 

4-76 

219797 

55 

6 

7 1 3098 

3-49 

932609 

1-27  1 

780489 

4-76 

21901 1 

54 

7 

7i33o8 

3-49 

93253l 

1-27 

780776 

4-76 

210225 

53 

8 

713517 

3-48 

932457 

1-27 

781060 

4-76 

2 1 8940 

52 

9 

713726 

3-48 

93238o 

1-27 

781346 

4-75 

218654 

5i 

10 

713935 

3-48 

9323o4 

1.27 

781631 

4-75 

218369 

5o 

ii 

9-7»4i44 

3-48 

9932228 

1-27 

9-781916 

4-75 

IO-2l8o84 

49 

13 

714353 

3-47 

932i5i 

1-27 

782201 

4*75 

217799 

48 

i3 

714561 

3-47 

932075 

1-28 

782486 

4-75 

217014 

47 

>4 

714760 

3-47 

931998 

1-28 

782771 

4-75 

217229 

46 

i5 

714970 

3-47 

931921 

1-28 

783o56 

4-75 

2 I 6944 

45 

16 

7i5i86 

3-47 

93 1 845 

1-28 

783341 

4-75 

2 I 6669 

44 

*7 

715394 

3-46 

931768 

1-28 

783626 

4-74 

216374 

43 

18 

7 1 56o3 

3-46 

931691 

1-28 

783910 

4-74 

2 1 6090 

43 

*9 

7 1 5809 

3-46 

931614 

1-28 

784195 

4-74 

2 i 58o5 

41 

20 

716017 

3-46 

93 1 537 

1-28 

784479 

4-74 

21 5521 

40 

21 

9-716224 

3-45 

9-93i46o 

1-28 

9-784764 

4-74 

io- 2i5236 

39 

22 

716432 

3-45 

93 1 383 

1-28 

785048 

4-74 

214952 

38 

23 

716639 

3-45 

93i3o6 

1-28 

785332 

4-73 

214668 

37 

24 

716846 

3-45 

931229 

I -29 

786616 

4-73 

214384 

36 

25 

717053 

3-45 

93 1 1 52 

I -29 

785900 

4-73 

214100 

35 

26 

717259 

3-44 

931075 

1-29 

786184 

4-73 

2i38i6 

34 

a7 

717466 

3 -44 

930998 

1 -29 

786468 

4-73 

2i3532 

33 

38 

717673 

3 • 44 

930921 

I -29 

786752 

4-73 

213248 

32 

*9 

7 1 7879 

3-44 

930843 

1-29 

787036 

4-73 

2 1 2964 

3i 

3o 

7 1 8o85 

3-43 

930766 

1-29 

787319 

4-72 

212681 

3o 

3i 

9-718291 

3-43 

9 • 93o688 

1-29 

9*787603 

4-72 

10-212397 

a9 

32 

718497 

3-43 

93o6i 1 

1-29 

787886 

4- 72 

212114 

28 

33 

718703 

3-43 

93o533 

1-29 

788170 

4-72 

2ii83o 

a7 

34 

718909 

3-43 

93o456 

1 -29 

788453 

4-72 

2 1 1 547 

26 

35 

719114 

3-42 

93o378 

1-29 

788736 

4-72 

211264 

25 

36 

719320 

3-42 

93o3oo 

1 -3o 

789019 

4*72 

210981 

24 

37 

719525 

3-42 

93o223 

i -3o 

789302 

4-71 

210698 

23 

38 

719730 

3-42 

93oi45 

1 -3o 

789585 

4-71 

2io4i5 

22 

39 

719935 

3-4i 

930067 

1 -3o 

789868 

4-71 

210132 

21 

4o 

720140 

3-41 

929989 

1 -3o 

7901 5i 

4-71 

209849 

20 

4i 

9-720345 

3-4i 

9-929911 

1 -3o 

9-790433 

4*71 

10-209567 

42 

720549 

3-4i 

929833 

1 -3o 

790716 

4-7* 

209284 

|8 

43 

720754 

3-4o 

929755 

i*3o 

790909 

4-71 

209001 

'7 

44 

720968 

• 3-4o 

929677 

1 -3o 

791281 

4-7* 

208719 

16 

45 

721162 

3-4o 

929599 

i-3o 

79 1 563 

4.7® 

208437 

i5 

46 

721 366 

3-4o 

929521 

1 -3o 

791846 

4-70 

208 I 54 

14 

4”» 

721670 

3-4o 

929442 

1 -3o 

792128 

4-70 

207872 

1 3 

48 

721774 

3-39 

929364 

i-3i 

792410 

4-70 

207590 

12 

49 

721918 

339 

929286 

i-3i 

792692 

4-70 

207308 

11 

5o 

j 722181 

339 

929207 

1 -3i 

792974 

4-7° 

207026 

10 

5i 

i 9-722385 

3-39 

9-929129 

i-3i 

9-793256 

4-70 

10-206744 

9 

53 

722588 

3-3o 

929060 

1 -3i 

793538 

4-69 

206462 

8 

53 

723791 

3-38 

928972 

i-3i 

7938i9 

4-69 

206181 

7 

54 

722994 

3-38 

928893 

i-3i 

794101 

4-69 

205899 

6 

55 

723197 

3-38 

928815 

i*3i 

794383 

4.69 

205617 

5 

56 

733400 

3-38 

928736 

1 -3i 

794664 

4.69 

205336 

4 

s 

7236o3 

3 - 37 

928657 

1 • 3 1 

794945 

4-69 

2o5o55 

3 

58 

7238o5 

3-37 

928678 

1 -3i 

795221 

4- 60 

204773 

2 

59 

724007 

3-37 

928499 

1 *3i 

7955o8 

4-68 

204492 

1 

60 

724210 

3-37 

928420 

1 -3i 

795789 

4-68 

20431 1 

0 

Cosine 

D. 

Sino 

D. 

1 Cotang. 

D. 

Tang. 

M. 

(58  DKGRKH8.) 


(32  DEGREES.)  A TABLE  OF  LOGARITHMIC 


DO 


M. 

Sine 

D. 

Cosine 

D.  , 

Tang. 

D. 

Cotang. 

| 

0 

9-724210 

3-37 

9-928420 

1-32  I 

9.795789 

4-68 

10-204211 

60  | 

i 

724412 

3-37 

928342 

1 -32  ' 

796070 

796361 

4-68 

2o393o 

5o  1 

a 

724614 

3-36 

928263 

1-32 

4-68 

203649 

58  | 

3 

724816 

3-36 

928183 

I -32 

796632 

4-68 

2o3368 

5? 

4 

725017 

3-36 

928104 

1-32 

796913 

4-68 

203087 

56 

5 

725219 

3-36 

928026 

I -32 

797*94 

4-68 

202806 

55  ; 

6 

725420 

3-35 

927046 

927867 

1-32 

797475 

797755 

798036 

4-68 

202525 

54 

2 

725622 

3-35 

1-32 

4-68 

202245 

53 

725823 

3-35 

927787 

927708 

927629 

I -32 

4-67 

201964 

52 

9 

726024 

3-35 

1-32 

7983 1 6 

4-67 

201684 

5i 

n 

726225 

3-36 

I -32 

798596 

4-67 

201404 

5o 

1 1 

9-726426 

3-34 

9-927649 

I -32 

9-798877 

799157 

4-67 

10-201 123 

49 

12 

726626 

3-34 

927470 

I -33 

4-67 

200843 

48 

i3 

726827 

3-34 

927390 

i*33 

799437 

4-67 

2oo563 

47 

14 

727027 

3-34 

9^27310 

i-33 

7997 >7 

4-67 

200283 

46 

i5 

727228 

3-34 

927231 

1 -33 

799997 

800277 

4-66 

200003 

45 

16 

727428 

3-33 

927151 

1 -33 

4-66 

199723 

44 

i ] 

727628 

3-33 

927071 

1 -33 

800567 

4-66 

1 99443 

43 

10 

727828 

3-33 

920991 

1 -33 

8oo83 6 

4-66 

199(64 

1 98884 

42 

>9 

728027 

3-33 

92691 1 

1 -33 

801 1 16 

4-66 

4* 

20 

728227 

3-33 

92683 1 

i-33 

801396 

4-66 

1 98604 

40 

21 

9-728427 

3-32 

9-926761 

1 -33 

9-801676 

4-66 

10- 198325 

3o 

22 

728626 

3-32 

926071 

1 -33 

801965 

4-66 

198045 

38 

23 

728825 

3-32 

926591 

1 -33 

8o2?34 

4-65 

i 197766 

37 

24 

729024 

3-32 

- 9265ii 

1 -34 

802613 

4-65 

197487 

197208 

36 

25 

729223 

3 • 3 1 

926431 

1 -34 

802792 

4-65 

35 

26 

—729422 

3 • 3 1 

92635i 

1 -34 

8o3072 

4-65 

196928 

196649 

34 

27 

729621 

3 • 3 1 

926270 

1 -34 

8o336i 

4-65 

33 

20 

729820 

3 • 3 1 

926190 

1 -34 

8o363o 

4-65 

196370 

32 

29 

730018 

3-3o 

926110 

i-34 

803908 

4-65 

196092 

3 1 

3o 

730216 

3-3o 

926029 

i-34 

804187 

4-65 

195813 

3o 

3i 

9 -73o4i 5 

3-3o 

9-925949 

925868 

,-34 

9 • 804466 

4-64 

10- 195534 

29 

32 

73o6 1 3 

3-3o 

i-34 

804745 

4-64 

195255 

28 

33 

73o8i 1 

3-3o 

925788 

1 -34 

8o5o23 

4-64 

*94977 

1 94698 

27 

34 

731009 

3-29 

926707 

925620 

1 -34 

8o53o2 

4-64 

26 

35 

731206 

3-29 

1-34 

80558o 

4-64 

194420 

25 

36 

73i4o4 

3-29 

925545 

i-35 

8o5859 

4-64 

194141 

24 

32 

731602 

3-29 

925465 

1 - 35 

806137 

4-64 

1 93863 

23 

38 

731799 

3-29 

925384 

i-35 

806416 

4-63 

193585 

22 

39 

731996 

3-28 

9253o3 

i-35 

806693 

4-63 

193307 

21 

40 

732193 

3-28 

925222 

i-35 

806971 

4-63 

193029 

20 

41 

3-28 

9925i4i 

i-35 

9-807249 

4-63 

io- 192751 

*9 

42 

3-28 

925o6o 

i-35 

807527 

807806 

8o8o83 

4-63 

192473 

l8 

43 

732784 

3-28 

924Q79 

1 -35 

4-63 

192195 

*7 

44 

732980 

3-27 

924897 

! -35 

4-63 

19*9*7 

16 

45 

733177 

3-27 

924816 

i-35 

8o836i 

4-63 

191639 

i5 

46 

733373 

3-27 

924735 

i-36 

8o8638 

4-62 

191362 

14 

47 

733560 

733766 

3-27 

924664 

1 - 36 

808916 

4 62 

191084 

1 3 

3-27 

924572 

i-36 

809193 

4-62 

1 90807 

1 1 

<9 

733961 

3-26 

92449 1 

1 -36 

80947* 

4-62 

190629 

1 1 

5o 

734157 

3-26 

924409 

1 - 36 

809748 

4-62 

190262 

10 

5i 

' 0-734353 

3-26 

9 924328 

i-36 

9810025 

4-62 

10-189976 

9 

52 

734549 

3-26 

924246 

1 - 36 

8io3o2 

4-62 

189698 

8 

53 

734744 

3-25 

924164 

1 -36 

8io58o 

4-62 

189420 

7 

54 

734939 

735i35 

3-25 

924083 

i-36 

810867 

4-62 

189143 

0 

55 

3-25 

924001 

1 -36 

81 1 1 34 

4-6i 

188866 

5 

56 

73533o 

3-25 

923919 

923837 

i-36 

81 1410 

4*61 

1 88590 

4 

5l 

735525 

3-25 

1 -36 

811687 

4-6i 

1 883 1 3 

3 

58 

735719 

3-24 

923755 

923673 

1-37 

81 1964 

4 - 6i 

1 88o36 

2 

59 

735914 

, 3*24 

1-37 

812241 

4-6i 

187760 

1 87483 

1 

60 

736109 

i 3-24 

923591 

1 -37 

812517 

4-6i 

0 

CoHino 

1 I). 

Sine 

D. 

Cotang. 

I). 

JL 

(57  DKOHKKR.) 


SINES  AND  TANGENTS.  (33  DEGREES.) 


51 


Sine 

D. 

Cosine 

D.  | 

Tang,  j 

D. 

Cotang. 

0 

9 -736100 
7363o5 

3-24 

9-923691 

1 -37 

9812517 

4-6i 

10-187482 

60 

1 

3-24 

923509 

1-37  ! 

812794 

4-6i 

181206 

186930 

5o 

2 

736498 

3-24 

923427 

923345 

1 -3t 

813070 

4 • 6 1 

58 

3 

736692 

736886 

3-23 

i-37 

8i3347 

4- 60 

186653 

57 

4 

3-23 

923263 

1-37 

8i3623  j 

4 -6c 

186377 

56 

5 , 

737080 

3-23 

923i8i 

1-37  I 

813890 

814176 

4- 60 

186101 

55 

6 1 

737274 

3-23 

923098 

i-37 

4 -6c 

185825 

54  , 

7 

737467 

3-23 

923016 

1-37 

814462 

4-  60 

185548 

53  1 

t 

737661 

3-22 

922033 

922851 

1-37 

814728 

4- 60 

186272  : 

52 

9 

737855 

738048 

3-22 

1 -37 

816004 

4-6o 

184996  j 

5i 

10 

3-22 

922768 

i-38 

816279 

4- 60 

184721  ; 

5o 

1 1 

9-738241 

3-22 

9-922686 

i-38 

9-81 5555 

4-59 

io- 184446  ; 

49 

12 

738434 

3-22 

922603 

i-38 

8 1 583 1 

4-69 

184169  ! 

48 

1 3 

738627 

8-21 

922520 

i-38 

816107 

4-69 

183896  i 

47 

i4 

738820 

3-21 

922438 

i-38 

8 1 6382 

4.59 

i 836 18  j 

46 

i5 

73qoi3 

3-21 

922355 

i-38 

8 i6658 

4-69 

183342 

45 

16 

739206 

3-21 

'922272 

922189 

i-38 

816933 

4-59 

183067 

44 

• 7 

739398 

3-21 

i-38 

817209 

4-59 

182791 

43 

1 8 

739590 

739783 

3-20 

922106 

i-38 

817484 

4-59 

182616  1 

42 

19 

3-20 

922023 

i-38 

817759 

8i6o36 

4-59 

182241  1 

4i 

30 

739975 

3-20 

921940 

i-38 

4-56 

181965 

4o 

21 

9-740167 

3-20 

9-921857 

1 39 

9-8i83io 

4-58 

10-181690 

3o 

22 

->4o359 

3-20 

921774 

1*39 

8i8585 

4-58 

1 8 1 4 1 5 

38 

23 

74o55o 

3-19 

921691 

1*39 

818860 

4-58 

181 140 

37 

24 

740742 

3-19 

921607 

1 *39 

8i9i35 

4-58 

i8o865 

36 

25 

740934 

3-19 

921524 

1-39 

819410 

4-58 

1 80590 

35 

26 

741125 

3-19 

921441 

1-39 

819684 

4-58 

i8o3i6 

34 

27 

741 3 16 

3*19 

921357 

1-39 

819959 

4-58 

180041 

33 

28 

741 5o8 

3*i8 

921274 

1 39 

820234 

4-58 

179766 

3s 

29 

741699 

3- 18 

921190 

1*39 

82o5o6 

4-67 

179492 

3i 

3o 

741889 

3- 18 

921107 

1 -39 

820783 

4.57 

1 792 1 7 

3o 

3i 

9-742080 

3*i8 

9-921023 

1*39 

9-821057 

4.57 

10-178943 

178668 

29 

32 

742271 

3- 18 

920039 

920856 

1 -4o 

82i332 

4.57 

28 

33 

742462 

3-17 

i-4o 

821606 

4.57 

178394 

27 

34 

742662 

3-17 

920772 

1 • 4o 

821880 

4.57 

178120 

26 

35 

742842 

3-17 

920668 

1 -4o 

822154 

4.57 

177846 

25 

36 

743o33 

3-17 

920604 

1 -4o 

822429 

822700 

4-67 

177571 

24 

37 

743223 

o'1! 

920520 

1 -4o 

4-5] 

27729 7 

177023 

23 

38 

7434i3 

3-i6 

920436 

1 -4o 

822977 

823260 

4-56 

22 

39 

743602 

3- 16 

920352 

1 -4o 

4-56 

176760 

21 

4o 

74379s 

3- 16 

920268 

1 -4o 

823524 

4-56 

176476 

20 

41 

9-743982 

3- 16 

9-920184 

1 -4o 

9-823798 

4-56 

io- 176202 

*9 

42 

43 

744171 

7443oi 

3-i6 

3 - 1 5 

920099 

920016 

1 -4o 

1 -4o 

824072 

824345 

4-56 

4-56 

175928 

i75655 

10 

17 

44 

74455o 

3 • 1 5 

919031 

919846 

1 -4i 

824619 

4-56 

i7538i 

16 

45 

74473o 

744928 

3 • 1 5 

1 *4i 

824890 

4-56 

175107 

i5 

46 

3 - 1 5 

919762 

1 *4i 

826166 

4-56 

174834 

14 

1? 

! $UZ 

3 • 1 5 

3- 14 

919677 

919593 

1 *4i 
i-4i 

825439 

8257i3 

4-55 

4-55 

174561 

174287 

i3 

12 

45 

745494 

3-14 

919508 

1 -4i 

825986 

4-55 

174014 

11 

5o 

' 745683 

3-i4 

919424 

1 -4i 

826259 

4-55 

173741 

10 

5i 

9-745871 
j 746069 
746248 

746436 

3-i4 

9-919339 

1 41 

9*826532 

4-55 

10- 173468 

9 

52 

3-i4 

919254 

1 -4i 

8268o5 

4-55 

176196 

8 

53 

54 

3 - 1 3 

4- 13 

919169 

919086 

1 *4i 

1 *4i 

827078 

827361 

4-55 

4-55 

172922 

172649 

172376 

2 

55 

746624 

3- 13 

919000 

i*4i 

827624 

4-55 

5 

56 

746812 

3- 13 

918915 

1-42 

82-7897 

826170 

4-54 

172103 

4 

57 

746909 

J 747J87 

3- 13 

918830 

1-42 

4-54 

171830 

3 

58 

3-u 

918745 

918669 

1-42 

828442 

4-54 

I7i558 

a 

J9 

747374 

747^62 

3-12 

I -42 

828715 

4-54 

171285 

1 

60 

3-12 

918574 

1-42 

828987 

4.54 

171013 

0 



| Coeino 

D. 

Sine 

D. 

Co  tang. 

D. 

Tang. 

M. 

18 


(56  DEGREES.) 


62 


(34  DEGREES.)  A TABLE  OF  LOGARITHMIC 


M. 

Siuo 

D. 

Connie 

D. 

Tang. 

D. 

Cotang. 

0 

9-747562 

3-ia 

9-918574 

918489 

1-42 

9-828987 

4*54 

lo- 171013 

60 

I 

! 747749 

3-12 

1-42 

829260 

4*54 

1 70740 

59  1 

a 

3-12 

918404 

142 

829532 

4*54 

170468 

58  ! 

3 

3 - 1.1 

9i83i8 

1-42 

829805 

4*54 

170195 

1 57 

4 

7483 10 

3 - 1 1 

918233 

1-42 

830077 

4*54 

169923 

1 6965 1 

56 

5 

748497 

748683 

3- 11 

918147 

1-42 

83o349 

4*53 

55  | 

6 

3 • 1 1 

918062 

1-42 

83o6ai 

4*53 

169879 

54  1 

l 

748870 

749006 

3*ii 

3io 

917076 

917891 

1 .43 

1 .43 

83o893 
83 1 i65 

4*53 

4*53 

169107 

168835 

53 

5a 

9 

749343 

3io 

917805 

1 .43 

831437 

4*53 

•68563 

5i 

10 

749429 

3io 

917719 

1 .43 

831709 

4*53 

168291 

5o 

ii 

9*749615 

3-io 

9-917634 

1 .43 

9 - 83 1 98 1 

4*53 

io- 168019 

49 

ta 

i3 

749801 

749987 

3io 

3 09 

917548 

917462 

1 .43 

i -43 

832253 

8325a5 

4-53 

4*53 

167747 

167476 

48 

47 

14 

750172 

3.09 

'917376 

1 .43 

832796 

4 53 

167204 

46 

i5 

75o358 

3*09 

917290 

1 .43 

833o68 

4*5a 

166932 

45 

16 

75o543 

3 09 

917204 

1 .43 

833339 

4*52 

166661 

44 

ll 

750729 

3 05 

917118 

1-44 

8336i 1 

4*5a 

166389 

43 

ib 

750914 

3- 08 

917032 

1-44 

833882 

4*5a 

1661 18 

42 

*9 

751099 

3.o8 

916946 

1*44 

834 1 54 

4*5a 

165846 

4i 

ao 

751284 

3.o8 

916859 

i*44 

834425 

4*5a 

165575 

4° 

9*751469 

75i654 

3.08 

3.08 

9.916773 

916687 

1*44 

1-44 

9 • 834696 

4*5a 

4*5a 

io* i653o4 
i65o33 

aa 

834967 

3? 

a3 

75i83o 

3 -08 

916600 

i*44 

835238 

4*5a 

164762 

37 

24 

752023 

3-07 

916614 

1*44 

8355o9 

4*5a 

164491 

36 

25 

752208 

3.07 

916427 

i*44 

835780 

4 • 5i 

164220 

35 

26 

752392 

3-07 

9i634i 

i*44 

836o5i 

4 * 5i 

163949 

1 63678 

34 

3J 

752546 

3-07 

916264 

i*44 

836322 

4-5i 

33 

2 a 

752760 

3-07 

916167 

1 -45 

836593 

4- 5i 

163407 

32 

39 

752944 

3>o6 

916081 

1-45 

836864 

4*5i 

i63 1 3o 

3i 

3o 

753128 

3.06 

916994 

1*45 

837134 

4- 5i 

162866 

3o 

3i 

9*7533ia 

3- 06 

9.915907 

1 .45 

9. 8374o5 

4*5i 

io- 162595 

29 

3a 

753495 

3 06 

915820 

1-45 

837675 

4- 5i 

i6a3a5 

28 

33 

34 

753679 

75386a 

3- 06 
3*o5 

918733 

915646 

i-45 

1-45 

837946 

838ai6 

4 - 5i 
4*5i 

162054 

161784 

2 

35 

754046 

3-o5 

9i5559 

1 .45 

838487 

4*5o 

i6i5i3 

25 

36 

754229 

3-o5 

915472 

9i5385 

1-45 

838757 

4*5o 

161243 

24 

37 

754412 

3-o5 

1-45 

839027 

4-5o 

160973 

23 

36 

39 

754595 

754778 

754960 

3o5 

3-04 

915297 

915210 

1*45 

1 .45 

839297 

839568 

4-5o 

4-5o 

160703 

160432 

22 

839838 

40 

3-o4 

91 5i a3 

1 .46 

4*5o 

160162 

20 

41 

9*755i43 

3 04 

9*9i5o35 

1 .46 

9*840108 

4-5o 

10-159892 

*9 

42 

755326 

3-04 

914048 

914860 

1 .46 

840378 

4*5o 

159622 

l8 

43 

7555o8 

3-o4 

1 .46 

840647 

4*5o 

1 59353 

17 

44 

755690 

3-04 

914773 

1 .46 

840917 

4.49 

1 59083 

16 

45 

755872 

3-o3 

914686 

1 .46 

841187 

4.49 

1 588! 3 

1 5 

46 

756o54 

3o3 

914598 

1 .46 

841457 

4.49 

158543 

14 

<7 

756236 

3-o3 

9i45io 

1 .46 

841726 

4.49 

158274 

i3 

4 6 

756418' 

3-o3 

914422 

1*46 

841996 

4.49 

i58oo4  1 

12 

49 

756600 

3o3 

914334 

1 .46 

842266 

4*49 

157734 

11 

5o 

756782 

3-02 

914246 

i*47 

842535 

4.49 

1 57465 

10 

5i 

9.756963 

3-02 

9914158 

i*47 

9 842805 

4.49 

10 • 1 57195 

9 

5a 

53 

757144 

767326 

3-02 

3*02 

914070 

9i3o8a 

913894 

i*47 

i*47 

843074 

843343 

4.49 

4.49 

156926 

156657 

8 

7 

54 

757507 
75768 8 

3-02 

i*47 

843612 

4-4Q 

156388 

0 

55 

3-oi 

913806 

i*47 

84388a 

4.48 

i56i 18 

5 

56 

5Z 

757869 

■j5bo5o 

3-oi 

3oi 

918718 

913630 

i*47 

i*47 

844 1 5 1 

844420 

4.48 

4*48 

1 55849 

1 5558o 

4 

3 

58 

758a3o 

3*oi 

918541 

i*47 

844689 

4.48 

1 553 1 1 

2 

844958 

59 

75841 1 

3-oi 

9i3453 

1*47 

4*48 

1 55o42 

1 

60 

75859i 

3oi 

9i3365 

1*47 

846227 

4.48 

1 54773 

0 

Comne 

D. 

Sine 

D. 

Ootnng. 

D. 

Tang.  1 

MJ 

(55  DKOKKB8.) 


SINES  AND  TANGENTS.  (35  DEGREES.; 


58 


KL 

Sin© 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

9-758591 

3oi 

9913365 

i-47 

9-845227 

4.48 

10-164773 

1 54504 
154236 

60 

I 

f 

?£$ 

3-oo 

3oo 

913276 

913107 

::3 

845490 

845764 

4.48 

4.48 

3 

3 

75gi32 

3oo 

913099 

1-48 

846o33 

4 48 

153967 

57 

4 

759312 

3-oo 

9i3oio 

1-48 

846302 

4.48 

153696 

56 

5 

75949a 

3-oo 

912922 

1 -48 

846570 

846809 

4-47 

i5343o 

55 

6 

299 

912833 

1 -48 

4-47 

i53i6i 

54 

7 

2-99 

915744 

1 -48 

847107 

847376 

4-47 

152893 

53 

8 

76003 1 

a 99 

912655 

1 -48 

4-47 

152624 

52 

9 

7602 1 1 

2-99 

912566 

1-48 

847644 

4-47 

1 52356 

5i 

10 

760390 

2 '99 

912477 

1-48 

847913 

4-47 

152087 

5o 

1 1 

9 -76o56o 

2 -98 

9*912388 

1 -48 

9-848181 

4-47 

io- 161819 

49 

12 

760748 

2-98 

912299 

1-49 

848449 

4-47 

i5i55i 

48 

13 

14 

760927 
761 106 

2-98 

2-98 

912210 

912121 

1-49 

1-49 

£3 

4-47 

4.47 

1 5i 283 

1 5 1 01 4 

s 

i5 

761285 

2-98 

9I2o3i 

1-49 

849254 

4-47 

1 50746 

45 

16 

761464 

2-98 

911942 

1-49 

849522 

4-47 

1 60478 

44 

*7 

761642 

2-97 

91 i853 

1-49 

849790 

85oo58 

4.46 

150210 

43 

1 8 

761821 

2-97 

911763 

1-49 

4-46 

149942 

149675 

43 

»9 

761999 

2-97 

9n674 

911584 

1-49 

85o325 

4-46 

41 

20 

762177 

2-97 

1 '49 

850693 

4.46 

149407 

40 

21 

q- 762356 

2 *97 

9'9ii495 

1 -49 

9-85o86i 

4.46 

10- 149139 
148871 

39 

22 

762534 

2-96 

9ii4o5 

1 -49 

85i 1 29 
85 1 396 

4.46 

38 

23 

762712 

762889 

2-96 

91 13 1 5 

1 -5o 

4-46 

148604 

37 

34 

2-96 

911226 

1 -5o 

85 1664 

4.46 

148336 

36 

25 

763067 

296 

9111 36 

1 -5o 

851931 

4.46 

148069 

35 

26 

763245 

2-96 

911046 

1 -5o 

862199 

4-46 

147801 

34 

37 

763422 

2-96 

910966 

910066 

1 -5o 

852466 

4-46 

147534 

33 

3$ 

763600 

2-95 

1 -5o 

852733 

4-45 

147267 

32 

8 

763777 

763964 

2-96 

2 95 

;:s£ 

1 -5o 

1 -5o 

853ooi 

853268 

4-45 

4-45 

146909 

146732 

3i 

3o 

3i 

9-764131 

2-95 

9-910596 

1 -5o 

9-853535 

4-45 

10- 146465 

30 

32 

764308 

2-95 

9io5o6 

1 -5o 

853802 

4-45 

146198 

28 

33 

764485 

2-94 

9io4i5 

1 -5o 

854069 

854336 

4-45 

145901 

145664 

37 

34 

764662 

2-94 

9io325 

1 -5i 

4*45 

26 

35 

764838 

2-94 

910235 

1 -5i 

8546o3 

4-45 

145397 

25 

36 

765o r 5 

2-94 

910144 

1 -5i 

854870 

4-45 

i45i3o 

24 

3-7 

766191 

2-94 

910054 

1 -5i 

855i3 7 

4-45 

144863 

23 

3$ 

765367 

2-94 

909963 

909873 

1 -5i 

8554o4 

4-45 

144596 

22 

39 

765544 

2-g3 

1 -5i 

855671 

8559o8 

4-44 

144329 

21 

4o 

765720 

2-93 

909782 

1 -5i 

4-44 

144062 

20 

4i 

9-765896 

2-93 

9-909691 

1 - 5 1 

9-866204 

4-44 

io- 143796 
143529 

42  j 

766072 

2-93 

909601 

1 -5i 

85647 1 

4-44 

l8 

43  | 

766247 

2-93 

9095 1 0 

1 -5i 

8567O7 

4-44 

143263 

>7 

44 

760423 

2-g3 

9094  IQ 

1 -5i 

867004  j 

4-44 

142996 

142760 

16 

45 

766698 

2-92 

909328 

1-52 

857270  i 
8575o7  1 

4-44 

1 5 

46 

766774 

2-92 

909237 

1-52 

4-44 

U2463 

14 

! 47 

766949 

2 92 

909146 

1-52 

867800 

4-44 

142197 

i3 

i 4$ 

767124 

2-92 

900055 

1-52 

858069 

4-44 

141901 

12 

! 49 

767300 

2-92 

908964 

1-52 

858336 

4-44 

141664 

1 1 

5c  ' 

767475 

2-91 

908873 

1-52 

858602 

4-43 

i4i3g8 

10 

5i 

9 767649 

2-91 

9-908781 

1-52 

9-858868 

4-43 

io- 141 x3a 

9 

52 

767824 

2 -91 

908090 

1 52 

859134 

4-43 

140866 

8 

53 

761990 

768173 

291 

908599 

1-52 

859400 

4-43 

1 40600 

7 

54 

2-91 

908507 

9084IO 

1-52 

859666 

4-43 

i4o334 

0 

55 

768348 

2-90 

1-53 

859932 

4-43 

1 40068 

5 

56 

768522 

2 90 

908324 

i-53 

860198 

860464 

4-43 

j 1 39802 

4 

52 

768697 

2 90 

908233 

i-53 

4-43 

j i39536 

3 

58 

768871 

2-90 

908UI 

1 -53 

860730 

4-43 

139270 

1 

8 

769045 

769219 

2-90 

2-90 

908040 

907968 

1 -53 
i -53 

860995 

861261 

4-43 

4-43 

1 39005 
138739 

1 

0 

Coeine 

D. 

1 Sino 

D. 

Cotang. 

D. 

Tang. 

M 

(54  DEGREES.) 


04 


(36  DEGREES.;  A TABLE  OF  LOGARITHMIC 


HaT 

Sine 

D 

Cosine 

| D. 

Tang. 

1 D> 

1 Cotang. 

. 

0 

» 

9-769210 

769393 

2-9° 

2-89 

9-907958 

907866 

1-53 

! 1 53 

9-861261 
861 527 

4-43 

4-43 

io- 138739 
138473 

60 

5y 

! 5 

769666 

2 89 

907774 

1 53 

861792 

4-42 

138208 

58 

3 

769740 

3-89 

907682 

i-53 

862058 

4-42 

137942 

57 

4 

769913 

3-89 

907690 

i-53 

862323 

4-42 

137677 

56 

5 

770087 

2-89 

907498 

i-53 

862589 

4-42 

137411 

55 

6 

770360 

2-88 

907406 

1 -53 

862854 

4-42 

137146 

54 

7 

770433 

2-88 

907314 

1 -54 

863119 

863385 

4-42 

1 3688 1 

53 

8 

770606 

2-88 

907222 

1 -54 

| 4-42 

1 366i 5 

53 

9 

770779 

770962 

3-88 

907 1 29 

1-54 

86365o 

! 4-42 

i3635o 

5i 

10 

2-88 

907037 

1-54 

863915 

4-42 

i36o85 

5o 

1 1 

9-771125 

2-88 

9-906945 

1 -54 

9-864180 

4-42 

10- 135820 

49 

13 

771298 

2-87 

906852 

1 -54 

864445 

4-42 

135555 

48 

i3 

771470 

2-87 

906760 

1 - 54 

864710 

4-42 

135290 

47 

14 

77«643 

2-87 

- 906667 

i-54 

864975 

4-4i 

i35o25 

46 

i5 

771816 

2-87 

906673 

906482 

1 -54 

865240 

4-4i 

134760 

45 

16 

77i987 

2-87 

1 • 54 

8655o5 

4.41 

134495 

134240 

44 

17 

772 1 59 

2 ■ 87 

906389 

i-55 

865770 

4-4i 

43 

18 

77233i 

2-86 

906296 

i-55 

866o35 

4-4i 

133965 

42 

*9 

7725o3 

2-86 

906204 

i-55 

8663oo 

4 • 4 1 

133700 

41 

30 

772675 

2-86 

906111  : 

i-55 

866564 

4-4i 

133436 

4o 

31 

9-772847 

2-86 

9-906018  , 

i-55 

9-866829 

4-4i 

io- 133171 

3g 

33 

773018 

2-86 

905925 

‘•55 

867094 

867358 

4-4i 

132906 

38 

33 

773190 

2-86 

905832 

1 • 55 

4-4i 

132642 

37 

24 

25 

77336i 

773533 

2-85 

2-85 

9o573o 

905645 

i-55 

i-55 

867623 

867887 

4.41 

4-4i 

132377 
k32 1 1 3 

36 

35 

26 

773704 

773875 

2-85 

9o5552  , 

i-55 

868 1 52 

4- 4o 

| 131848 

34 

37 

2-85 

9o5459 

i-55 

868416 

4-4o 

1 3 1 584 

33 

28 

774046 

2-85 

9o5366 

i-56 

868680 

4.40 

i3i32o 

32 

39 

7742n 

2-85 

905272 

1 - 56 

868945 

4- 4o 

i3io55 

3i 

3o 

774388 

2-84 

905179 

i-56 

869209 

4-4o 

( 130794 

3o 

3i 

9-774558 

2-84 

9-9o5o85 

i-56 

9'86$7 

87OOOI 

4-4o 

io- 130527 

39 

33 

33 

774729 

774899 

2-84 

2-84 

904992 

904898 

i-56 

i-56 

4.40 

4.40 

1 3o263 
129990 

129735 

28 

27 

34 

775070 

2-84 

904804 

i-56 

870265 

4-4o 

26 

25 

35 

775240 

2-84 

904711 

i-56 

870520 

87O7QJ 

87IO57 

4.40 

129471 

. 36 
31 

7754io 

77558o 

2-83 

2-83 

904617 

904623 

i-56 

1 -56 

4.40 

4-4o 

129207 

128945 

24 

23 

38 

775750 

2-83 

904429 

904335 

‘•57 

87l32I 

4.40 

128679 

1 28416 

22 

39 

776920 

2-83 

1 * 57 

87l585 

4-4o 

21 

40 

776090 

2-83 

904241 

i-57 

87 I 849 

4.39 

1 281 5i 

20 

4i 

9-776259 

2-83 

9-904I47 

i-57 

9-8721 12 

4.39 

10-127888 

42 

43 

776429 

776698 

2-82 

2-82 

9o4o53  | 
qo3q5q  ! 

i-57  1 
i-57 

872376 

872640 

! 4-39 

4-39 

127624 

127360 

I§ 

n 

9o3?64 

44 

776768 

2-82 

i-57 

872903 

4-39 

127097 

126833 

16 

45 

776937 

777106 

2-82 

903770 

1-57 

873167 

4-39 

i5  | 

46 

2-82 

903676 

i-57 

873430 

4-39 

126570 

i4 

47 

777275 

2 • 8 1 | 

9o358i 

i-57 

873694 

873967 

4.39 

i263o6  | 

i3 

i 48  I 

777444 

2 • 81 

903487 

‘•57 

4-39 

126043 

12 

i 49 

777613 

2 • 8l 

903392 

i-58 

874220 

4.39 

125780 

11 

5o 

777781 

1-81 

903298 

i-58 

874484 

4-39 

i255i6 

10 

5i 

9 777950 
778119 

2-81 

99o32o3 

i-58 

9-874747 

4-39 

10  125253 

9 

5'. 

2*81 

903108 

i-58 

875010 

- 4-3o 

124990 

8 

53 

54 

778287 

778455 

2 -8o 

2 -8o 

9o3oi4  1 
902019  | 
902824 

1 -58 
i-58 

876273 

875536 

4-38 

4-38 

124727 

1 24464 

I 

55 

778624 

2 • 80 

i-58 

876800 

4-38 

124200 

5 

56 

778702 

778960 

2 • 80 

902529 

i-58 

876063 

4-38 

123937 

4 

*7 

2 • 80 

902634 

i-58 

876326 

4-38 

123674 

3 

58 

779138 

2 -8o 

903539 

i-59 

876589 

4-38 

123411 

2 

59 

779295 

779463 

2 -->9 

903444 

i-59 

876851 

4-38 

1 23 149 

1 

60 

i-79 

902349  | 

i-59 

8771U 

1 4-38 

122886 

0 

Gosino 

1 I). 

Sine 

D. 

Ootang. 

rm 

Tang 

M. 

(53  DEURBB8.) 


SINES  AND  TANGENTS.  (37  DEGREES.; 


50 


M. 

Sine 

D. 

Cosino 

D. 

Tang. 

D. 

Cotang. 

0 

0-779463 

2-79 

9-902340 

902253 

1*59 

1 9-8771 14 

4-38 

10-122886 

60 

i 

77963 1 

2-79 

1-59 

I 877377 

4-38 

122623 

5o 

a 

779708 

779 906 
780133 

2-79 

902 1 58 

1*59 

877640 

4-38 

122360 

58 

3 

4 

2-79 

a-79 

902063 

901967 

1.59 

1-59 

4-38 

4-38 

122097 

121835 

£ 

5 

78o3oo 

2-78 

901872 

‘•59 

878428 

4-38 

121572 

55 

6 

7 

780467 

780634 

2.78 

2-78 

901776 

901681 

1-59 

1-59 

878691 

878953 

4-38 

4-37 

121309 

1 2 1047 

54 

53 

6 

780801 

2-78 

90 1 585 

1 -59 

879216 

4-37 

120784 

5 a 

9 

780968 

2-78 

90 1 490 

i-59 

879478 

4-37 

120522 

5i 

10 

781 1 34 

2-78 

901394 

1 -6o 

879741 

4-37 

120259 

5o 

li 

q-78i3oi 

2-77 

9-901298 

1 -6o 

9-88ooo3 

4-37 

10-119907 

119735 

4o 

ia 

781468 

2-77 

901202 

i-6o 

880265 

4-37 

48 

i3 

781634 

2-77 

901 106 

1 -6o 

88o528 

4-37 

II9472 

47 

U 

i5 

781800 

781966 

2-77 

2-77 

901010 

900914 

coo8 1 8 

1 -6o 

1 -6o 

880790 

881002 

4-37 

4-37 

1 I921O 

I I 8948 

46 

45 

16 

782132 

2-77 

1 -6o 

88 1 3 1 4 l 

4-37 

i i 8686 

44 

l7 

782298 

2-76 

900722 

, -6o 

1 881576 

4-37 

11S424 

43 

18 

782464 

2-76 

900626 

1 -6o 

881839 

4-37 

118161 

42 

>9 

782630 

2-76 

900529 

900433 

1 -6o 

882101 

4-3-7 

117809 

4i 

ao 

782796 

2-76 

1 -6i 

882363 

4-36 

j 117687 

40 

ai 

9-782961 

2-76 

9-900337 

1 -6i 

9-882625 

4-36 

io- 1 17375 

3g 

aa 

783127 

2-76 

900240 

1 -6i 

882887 

4-36 

117113 

38 

a3 

783292 

783458 

3.75 

900144 

1 -6i 

883148 

4-36 

ii6852 

3I 

24 

2-75 

900047 

1 -6i 

8834io 

4-36 

1 16590 

36 

a5 

a6 

27 

783623 

783788 

783953 

2-75 

2-75 

89995 1 

1 -6i 

1 -6i 

1 -6i 

883672 

883934 

884196 

884407 

4-36 

4-36 

4-36 

1 16328 

1 16066 

1 1 58o4 

35 

34 

33 

899767 

30 

7841 18 

2.75 

899660 

1 • 61 

4-36 

ii5543 

3a 

29 

784282 

2-74 

899564 

i-6i 

884719 

4-36 

1 i 5281 

3i 

3o 

784447 

2.74 

899467 

1 -6a 

884980 

4-36 

1 1 5o20 

3o 

3i 

9-784612 

a -74 

9 • 899370 

1 -6a 

9-885242 

4-36 

IO- I 14758 

29 

3a 

784776 

2-74 

899273 

1 -6a 

8855o3 

4-36 

114407 

28 

33 

784941 

2-74 

899176 

1 -6a 

885765 

4-36 

114235 

27 

34 

785io5 

! 2-74 

899078 
89898 1 

898884 

1 -62 

886026 

4-36 

1 13974 

26 

35 

36  ! 

785269 

785433 

2-73 

1 -62 

1 62 

886288 

886549 

4-36 

4-35 

1 13712 

1 1 345i 

25 

24 

785597 

78576, 

2-73 

898787 

1 • 62 

886810 

4-35 

113190 

23 

36 

2-73 

898689 

1 -62 

887072 

887333 

4-35 

112928 

22 

*9 

785925 

2-73 

898592 

, -6a 

4-35 

1 1 2667 

21 

40 

786089 

2-73 

898494 

i-63 

887594 

4-35 

1 1 2400 

20 

41 

9-786252 

2-72 

9-898397 

; 1 -63 

9 -887855 
8881 16 

4-35 

IO- 112145 

*9 

42 

786416 

2-72 

898299 

i-63 

4-35 

1 I 1884 

|8 

43 

786579 

j 2-73 

898202 

1 -63 

888377 

888639 

4-35 

I I 1623 

*7 

44 

786742 

j 2-73 

! 898104 

1 -63 

4-35 

1 i i36i 

16 

45 

786906 

| 2-72 

898006 

1 -63 

888900 

4-35 

11 1100 

i5 

46 

787069 

3-72 

897908 

897810 

i-63 

, 889160 

4-35 

1 10840 

14 

47 

787232 

3-71 

1 63 

889421 

4-35 

1 10579 

i3 

4b 

$$ 

1 2-71 

897715 

897614 

1 -63 

889682 

4-35 

iio3i8 

■3 

49 

2-71 

i-63 

889943 

4-35 

iioo5t 

1 1 

5o 

787720 

a-71 

897516 

1 -63 

890204 

4 34 

•09796 

10 

5i 

9.787883 

788045 

a • 7* 

9-897418 

i -64 

9 • 890465 

4-34 

jio  109535 

9 

5a 

2-71 

897320 

1 -64 

890725 

4-34 

109275 

8 

53 

788208 

271 

897223 

1 -64 

890986 

4-34 

1 090 1 4 

108753 

7 

54 

788370 

78853a 

788694 

788856 

2-70 

897123 

1 -64 

891247 

4 • 34 

6 

55 

56 

2-70 

2 70 

897025 

896926 

1 -64 

1 -64 

891507 

891768 

4-34 

4 • 34 

io84o3 

108232 

5 

4 

1 3.70 

896828 

1 -64 

892028 

4*34 

i 107972 

3 

58 

789018 

2-70 

896729 

896631 

1 64 

892289 

4 - 34 

! 10771 1 

a 

59 

789180 

a * 70 

1 1 -64 

892549 

4*34 

107451 

1 

60 

789342 

: 2-69 

896532 

1 -64 

892810 

4*34 

107190 

0 

Coaine 

1 D. 

Sine 

1 D. 

Cotang. 

D. 

1 Tang. 

M. 

(f»2  DRGRRB8.) 


56 


(38  DEGREES.)  A TABLE  OP  LOGARITHMIC 


M. 

Sino 

D. 

Cosine 

D. 

Tang. 

D. 

CoUng. 

0 

9-789343 

2-69 

9-896532 

1-64 

9-892810 

4-34 

io- 107100 

60 

i 

3 

789504 

789665 

*•*9 

169 

896433 

896335 

1 -65 

1 -65 

893070 

893361 

4-34 

4-34 

106080 

106669 

is 

3 

i 

789837 

789988 

2-69 

2 -69 

896236 

896137 

896038 

1 -65 

1 -65 

893591 

893861 

4-34 

4 34 

106409 

106149 

iz 

5 

790149 

2-69 

1 -65 

8941 1 1 

4 * 34 

105889 

55 

« 6 
l 

7903 1 0 
790471 
790633 
790703 
790954 

2-68 

2-68 

895939 

895840 

i-65 

i-65 

904371 

8946J2 

4-34 

4-33 

105629 

io536r 

54 

53 

8 

9 

2-68 

2-68 

896741 

895641 

i-65 

i-65 

894892 

895162 

4-33 

4 33 

io5io8 

104848 

52 

5i 

!C 

2-68 

895542 

i-65 

896412 

4-33 

104588 

5o 

1 I 

9-791 1 1 5 

2-68 

9-895443 

i-66 

9 -896672 
896982 

4-33 

10-104328 

12 

791275 

791486 

2-67 

895343 

1-66 

! 4-33 

104068 

|3 

2-67 

896244 

1-66 

896192 

896462 

1 4 33 

1 o38o8 

47 

>4 

791696 

79*707 

2-67 

895145 

1-66 

4 33 

io3548 

46 

i5 

2-67 

896045 

i-66 

8967 i 2 

4-33 

103288 

45 

16 

79*9*7 

2-67 

894945 

894846 

i-66 

89697 1 

4-33 

103029 

44 

'l 

792077 

792237 

2-67 

2-66 

i-66 

897281 

4-33 

102769 

102609 

43 

18 

894746 

i-66 

897491 

897761 

898010 

4-33 

43 

19 

792307 

792557 

2-66 

894646 

i-66 

4-33 

102249 

41 

30 

2-66 

894546 

i-66 

4-33 

101990 

4o 

31 

9.7927*6 

792876 

3-66 

9-894446 

1-67 

9-898270 

4-33 

io- 101730 

12 

22 

2-66 

894346 

1-67 

898580 

4-33 

101470 

23 

793o35 

2-66 

894246 

1-67 

8987.89 

4-33 

10121 1 

37 1 

34 

j5 

793195 

793354 

2-65 

2-65 

894146 

894046 

1-67 

1-67 

899049 

899308 

4-32 

4-32 

100961 

100692 

100482 

36 

35 

26 

7935i4 

2-65 

893946 

893846 

1-67 

899568 

4-32 

34 

ll 

793673 

2-65 

1-67 

899827 

4-32 

100173 

33 

793832 

2-65 

893745 

1-67 

900086 

4-32 

099914 

099654 

32 

*9 

79399* 

2-65 

893645 

1-67 

900346 

4-32 

3i 

3o 

794100 

2-64 

893544 

1 -67 

900606 

4-32 

099396 

3o 

3i 

9 • 794308 

2-64 

9 • 893444 

i-68 

9 • 900864 

4-32 

10-099136 

098876 

!2 

32 

794467 

2-64 

893343 

1-68 

->  901124 

4-32 

33 

794626 

2-64 

893243 

i-68 

90 1 383 

4-32 

098617 

098360 

a7 

34 

794784 

2 64 

893142 

i-68 

901642 

4-32 

26 

35 

794942 

2-64 

893041 

i-68 

901901 

4-32 

098099 

25 

36 

796501 

2-64 

892940 

892839 

i-68 

902160 

4-32 

097840 

34 

h 

795259 

2-63 

i-68 

902419 

4-32 

097581 

33 

38 

39 

795417 

795576 

795733 

2-63 

2-63 

892739 

892638 

i-68 

i-68 

902679 

902938 

4-32 

4-32 

097321 

097062 

32 

31 

40 

2-63 

892536 

i-68 

903197 

4 - 3 1 

096803 

20 

41 

9-795891 

2-63 

9-892435 

1 -69 

9-9o3455 

4 - 3i 

10-096545 

12 

42 

796049 

2-63 

892334 

1 69 

903714 

4 - 3 1 

096286 

43 

796206 

2-63 

892233 

1 -69 

903973 

4 - 3 1 

096027 

095768 

095609 

*7 

44 

796364 

2-62 

892132 

1 -69 

904282 

4 • 3 1 

16 

i5 

45 

796621 

2-62 

892630 

1 69 

904491 

4 • 3 1 1 

46  | 

796679 

2-62 

te 

891726 

1 -69 

904760 

4-3.  | 

095260 

14 

u\ 

796836 

262 

1 -69 

906008 

4 - 3 1 1 

094993 

094733 

1 3 

796993 

797160 

2-62 

1 -69 

905267 

4 * 3 1 ! 

12 

49 

2-61 

891624 

i -69 

905526 

4-3i  j 

094474 

1 1 

60 

797307 

2 • 61 

891,523 

1.70 

905784 

4-3i 

094216  I 

10 

5i 

9 797464 
797621 

2 • 6 1 

9-891421 

• •70 

9-906043 

4 - 3 1 

10-093957  j 
093698 

9 

53 

2 • 6 1 

891319 

1-70 

906302 

4 • 3 1 j 

8 

53 

54 

797777 

797934 

798091  . 

798247 

79840,3 

798560 

798716 

798871 

2-61 

2-61 

891217 

89  in  5 

1-70 

1-70 

9o656o 

906819 

4 • 3 1 1 

4 • 3 1 

093440 

093181 

l 

55 

5* 

ll  \ 

69  j 

2 • 61 
2-61 

2- 60 

891013 
89091 1 
890809 

1-70 

1.70 

170 

907594 

9078.62 

908 1 1 1 

4 - 3 1 

4 • 3 1 

4 - 3 1 

092923 

092664 

092406 

5 

4 

3 

1 -6o 

3 -6o 

890707 

890606 

170 

1-70 

4 - 3 1 
4-3o 

092148 
091889  j 

3 

1 

60  1 

2 • 60 

890503 

1.70 

908369 

4-3o 

091631 

0 

Cosine 

D.  _ 

Sine 

P.  1 

Cotang. 

I). 

Tong.  i 

JVL 

(51  DRORKKK.) 


SINES  AND  TANGENTS.  (39  DEGREES., 


67 


Sine 

D. 

Cosine 

1 D* 

Tang. 

D. 

Cotftng. 

0 

9-798872 

2- 60 

9 • 890503 

1 -70 

9-908369 

4*3o 

10  091631 

60 

I 

799028 

2- 60 

890400 

1.71 

908628 

4-3o 

091372 

59 

2 

799184 

2 • 60 

890198 

i*7* 

908886 

4-3o 

091114 

58 

3 

79933o 

2 69 

890196 

1 *7* 

909144 

4-3o 

090856 

57 

4 

799495 

799651 

2-69 

890093 

171 

909402 

4-3o 

090598 

56 

5 

2-59 

889990 

889888 

* *7* 

909660 

4-3o 

090340 

55 

6 

799806 

2-69 

* *7* 

909918 

4-3o 

390082 

54 

l 

199962 

800117 

2-59 

2-59 

889785 

889685 

» *7* 

* -7i 

91 °i 77 
910435 

4-3o 

4-3o 

089823 

o89565 

53 

52 

9 

800272 

2-58 

889579 

1 *7* 

910693 

9 1 090 1 

4-3o 

089307 

5i 

10 

800427 

2-58 

889477 

i-7* 

4-3o 

389049 

5o 

ii 

9-8oo582 

2-58 

9-889874 

1 .72 

9-911209 

4-3o 

10-088791 

o88533 

49 

13 

800737 

800892 

2-58 

889271 

1-72 

911467 

4-3o 

48 

i3 

2-58 

889168 

1-72 

91 >724 

4-3o 

088276 

47 

14 

801047 

2-58 

889064 

1-73 

911982 

4-3o 

088018 

46 

2-58 

1 -72 

912240 

4 • vJO 

087760 

087602 

43 

16 

80 1 356 

2-57 

888858 

1 -72 

912498 

4-3o 

44 

8oi5i 1 

2-57 

2-57 

888755 

1-72 

912736 

9i3oi4 

4-3o 

4-29 

087244 

43 

42 

801819 

1-72 

000900 

>9 

2-67 

888548 

1-72 

913271 

4-29 

086729 

41 

20 

801970 

2*57 

888444 

1-73 

913529 

4-29 

086471 

40 

31 

9-802128 

2-57 

9 • 88834 1 

1-73 

9-913787 

4*29 

10  086213 

3g 

32 

802282 

2-56 

888237 

1-73 

914044 

4-29 

085956 

38 

23 

802436 

2-56 

888i34 

i-73 

9i43o2 

4-29 

085698 

37 

24 

802 58o 

2-56 

888o3o 

i-73 

914660 

4-29 

085440 

36 

25 

802740 

2-56 

887926 

1-73 

914817 

915076 

915332 

4-29 

o85i83 

35 

26 

802897 

2-56 

887822 

i-73 

4-29 

084925 

34 

37 

8o3o5o 

2-56 

887718 

1-73 

4-29 

084668 

33 

2o 

803204 

2-56 

887614 

1-73 

915590 

4*29 

084410 

32 

39 

8o3357 

3-55 

887510  1 

i-73 

915847 

4-29 

o84i53 

3i 

3o 

8o35ii 

2-55 

887406 

1-74 

916104 

4-29 

083896 

3o 

3i 

9 • 803664 

2-55 

9-887302 

1-74 

9-916362 

4-29 

io-o83638 

29 

32 

803817 

2-55 

887198 

1-74 

916619 

4-29 

o8338i 

28 

33 

803970 

2-55 

887093 

886989 

886885 

i-74 

916877 

4-29 

o83 1 23 

27 

34 

804123 

2-55 

1-74 

9i7i34 

4.29 

082866 

26 

35 

804276 

2-54 

1-74 

917391 

4.29 

082609 

25 

36 

804428 

2-54 

886780 

1-74 

917648 

4-29 

082352 

24 

h 

804681 

2-64 

886676  ; 

1-74 

917905 

4-29 

082096 

23 

38 

39 

. 804734 

2-54 

886571  I 

1-74 

9i8i63 

4-28 

081837 

22 

804886 

2-54 

886466  ; 

1-74 

918420 

4.28 

o8i58o 

21 

4o 

* 8o5o39 

2-64 

88636s  | 

1 *75 

918677 

4-28 

o8i323 

20 

4i 

9-806191 

2-54 

9-886267  i 

1-75 

9-918934 

4-28 

10 -081066 

*2 

42 

8o5343 

2-53 

886162  ! 

1 *75 

919191 

4-28 

080809 

18 

43 

805496 

2-53 

886047 

1-75 

919448 

4-28 

o8o552 

>7 

44 

8o5647 

2-53 

885942 

1-75 

919705 

4-28 

080295 

080038 

16 

45 

806799 

806951 

2-53 

885837 

i-75 

919962 

4-28 

i5 

46 

2-53 

885732  1 

i-75 

920219 

4-28 

079781 

14 

47 

806 1 o3  1 

2-53 

885627 

1-76 

920476 

4-28  1 

079624 

1 3 

46 

806264 

2-53 

885522 

1-75 

920733 

4-28 

079267 

12 

49 

806406 

2-52 

885416 

i-75 

920990 

4-28 

079010 

1 1 

5o 

8o6557 

2-52 

8853 1 1 

1-76 

921247 

4-28 

078753 

10 

5 1 

9 806709 

2-52 

9-885205 

1-76 

9-92i5o3 

4-28 

1 0 • 078497 

9 

52 

806860 

2-52 

885ioo 

i-76 

921760 

4-28 

078240 

8 

53 

80701 1 

2-52 

884994 

884889 

1-76 

922017 

4-28 

077983 

7 

54 

807163 

2-52 

1-76 

922274 

9225jo 

4.28 

077726 

0 

55 

807314 

2-52 

884783 

1 -76 

4-28 

077470 

5 

56 

807465 

2 • 5 1 

884677 

1-76 

922787 

4-28 

07ju3 

4 

807615  1 

2 - 5l 

884572 

.•76 

923644 

4-28 

070956 

3 

58 

807766 

2 - 5l 

884466 

1 -76 

9233oo 

4-28 

076700 

2 

& 

80^67 

2-5i 

2 - 5 1 

88436o 

884354 

1-76 

>•77 

923557 

9238i3 

4-27 

4-27 

076443 

076187 

0 

Coeino  1 

D. 

Sino 

D.  ! 

Cotiing. 

D. 

Tang. 

M. 

(60  DBOH1SK8.) 


(40  DEGREES.)  A TABLE  OF  LOGARITHMIC 


58 


M 

81110 

1 D* 

j Cosmo 

1 D. 

| Tang. 

j D. 

-1 - 

Cotang. 

0 

9 • 808067 

a - 5i 

9-884254 

'•77 

9*9238i3 

427 

10076187 

60 

i 

808218 

2 • 5 1 

884148 

'•77 

924070 

4-27 

075930 

69 

2 

8o8368 

a-5i 

884042 

'•77 

924327 

924583 

427 

075673 

58 

3 

8o85 1 9 

2-5o 

883936 

1 883829 

'•77 

4-2T 

1 075417 

i 57 

4 

808669 

2 -5o 

'•77 

924840 

4-27 

l 075160 

5<! 

5 

808819 

a-5o 

883723 

'•77 

925096 

925352 

4-27 

074904 

55 

6 

808969 

! 2 -5o 

I 883617 

1 -77 

427 

1 074648 

54 

7 

8091 19 

a-5o 

8835io 

'•77 

025609 

925865 

4-27 

074391  1 53 

1 074135  5a 

8 

809269 

a-5o 

883404 

, ' -77 

4-27 

9 

8094 1 9 

249 

883297 

! 1.78 

926122 

j 4-27 

073878 

5i 

IC 

809569 

a 49 

883 1 9 1 

1 78 

926378 

4-27 

07362a 

5o 

ii 

9 • 8097 1 8 
809868 

I 2 49 

9- 883o84 

1-78 

9-926634 

4-27 

10073366 

49 

ia 

2-49 

882077 

882871 

1 -78 

926890 

4-27 

0731 10 

48 

i3 

810017 

2-49 

1-78 

927'47 

4-27 

072853 

47 

i4 

810167 

2-49 

882764 

.■78 

927403 

4-27 

072597 

46 

i5 

8 1 o3  j 6 

2-48 

882657 

i-78 

927659 

4-27 

072341 

45 

16 

8 1 0465 

2-48 

88255o 

.•78 

927915 

4-27 

072085 

44 

»7 

810614 

2-48 

882443 

.78 

928171 

4-27 

071820 

071573 

43 

ro 

810763 

2.48 

882336 

! • - 79 

928427 

4-27 

42 

*9 

810913 

2-48 

882229 

; '-79 

928683 

4-27 

071317 

4i 

ao 

81 1061 

2-48 

882121 

' -79 

928940 

4-27 

07 1 060 

40 

ai 

981 1210 

2-48 

9-882014 

'•79 

9-929196 

929452 

4-27 

1 0 • 070804 

3q 

23 

81 1 358 

2-47 

881907 

! ' 79 

4-27 

070548 

38 

a3 

8 1 1 5o7 

2-47 

881799 

881692 

'•79 

929708 

4-27 

070292 

37 

24 

81 1 655 

2-47 

>•79 

929964 

4-26 

070036 

36 

35 

8 i 1 804 

2-47 

88 1 584 

'•79 

930220 

4-26 

009780 

35 

36 

81 1952 

2-47 

881477 

'•79 

93o475 
93o73 1 

4 -26 

069625 

34 

*7 

812100 

2-47 

881 369 

1.79 

4-26 

069260 

069013 

068757 

068001 

33 

28 

812248 

2-47 

881261 

1 -8o 

930987 

93i243 

4*26 

32 

29 

81 2396 

2-46 

881 1 53 

1 -8o 

4-26 

3i 

3o 

8ia544 

2-46 

881046 

1 -8o 

93i499 

4-26 

3o 

3i 

9-812692 

2-46 

9-880938 

1 -8o 

9-931755 

4-26 

io- 068245 

29 

3a 

812840 

2-46 

88o83o 

1 -8o 

932010 

4-26 

067990 

067734 

28 

33 

812988 

2-46 

880722 

1 -8o 

932266 

4-26 

37 

34 

8 1 3 1 35 

2-46 

88o6i3 

1 -8o 

932522 

4-26 

067478 

26 

35 

8 1 3 283 

2-46 

88o5o5 

1 -8o 

932778 

933o33 

4-26 

067222 

25 

36 

8i343o 

2-45 

880397 

1 -8o 

4-26 

066967 

24 

37 

813578 

2-45 

880289 

1 -81 

933289 

933545 

4-26 

06671 1 

23 

3$ 

8 1 3725 

2-45 

880180 

1 -81 

4-26 

o66455 

23 

3q 

813872 

2-45 

880072 

1 -81 

9338oo 

4-26 

066200 

2 1 

4o 

814019 

2-45 

879963 

1 -8i 

934o56 

4-26 

065944 

20 

41 

9-8141 66 

2-45 

9 -879855 

1 -81 

9-9343ii 

4-26 

10-066689 

065433 

*9 

43 

8 1 43 1 3 

2-45 

879746 

i -81 

934567  | 
934823  j 

4-26 

18 

43  1 

81  1460 

2-44 

879637 

1 -8i 

4-26 

065177 

'7 

44 

8 1 4607 

2-44 

879529 

1 -8i 

935078 

935333 

4-26 

064922 

16 

45  ; 

814753 

3-44  1 

879420 

1 -81 

4-26 

064667 

i5 

4t 

8 1 4900 

2-44 

8793 1 1 

1 -8i 

935589 

4-26 

064411 

14 

s 

49 

81 5o46 

2-44 

879202 

1 -82 

935844  j 

4-26 

064166 

1 3 

815193 

8i5339 

2-44 

2-44 

879093 

878984 

1-82 

1 82 

936100  1 
936355  i 

4-26 

4-26 

063900 

o63645 

12 

5o 

8 15485 

2-43 

878875 

1 -82 

936610 

4-26 

063390 

0 

5i 

981 563 1 

2-43 

9-878766 

878656 

1 -8a 

9. 936866  1 

4-a5 

1 0 o63 1 34 

9 

5a 

815778 

2-43 

1 -8a 

937  m 1 

4-a5  I 

062879 

8 

53 

54 

815924 

816069 

2-43 

2-43 

878547 

878438 

1-82 

1-82 

937376  ; 
937632 

4-25 
4-a5  1 

062624 

o62368 

l 

55 

816215 

2-43 

878328 

1-82 

937887 

930142 

938398 

4-a5  ! 

0621 i3 

5 

56 

8i636i 

2-43 

878219  j 

i-83 

4-25 

061 858 

4 

5i 

816607 

242 

878109 

1 -83 

4-25 

06160? 

3 

I 

938653 

58 

8 1 665a 

a -42 

877099 

877890 

i-83 

4-25  , 

o6i347 

a 

59 

816798 

2-42 

1 -83 

938908 

4-25 

061092 
060837  ! 

1 

6c 

8 1 6943 

2-42 

877780 

1 83 

939163  I 

4-25 

0 

Coftino 

D. 

Sino 

D. 

Cotang.  | 

D.  1 

Tang.  | 

M. 

(49  DEGREES.) 


SINKS  AND  TANGENTS.  (41  DEGREES.; 


rM.~ 

Sine 

D. 

Cosine 

i D* 

Tang. 

D. 

Cotang. 

0 

9-816943 

2-42 

9-877780 

877670 

i-83 

9-939163 

4-25 

10-060837 

60 

i 

817088 

2-42 

i-83 

939418 

4-25 

o6o582 

59 

2 

817233 

2-42 

877560 

i-83 

939673 

4-25 

060327 

58 

3 

817379 

2-42 

8',  745o 

i-83 

939928 

4-25 

060072 

57 

4 

817524 

2-41 

8-(  7340 

i-83 

940183 

4-25 

069817 

56 

5 

817668 

2-41 

87  723c 

1-84 

940438 

4-25 

069562 

55 

6 

817813 

2-41 

87  7120 

1-84 

940694 

4-25 

059306 

54  1 

7 

817958 

2-41 

1]IZ 

1-84 

940949 

4-25 

059051 

53 

6 

8i8io3 

2-41 

1-84 

941204 

4-25 

058796 

058542 

52 

9 

818247 

2-41 

876789 

1-84 

941458 

4-25 

5i 

IO 

818392 

2-41 

876678 

1-84 

941714 

4-25 

058286 

5o 

ii 

9-8i8536 

2-40 

9-876568 

1-84 

9-941968 

4-25 

io-o58o32 

49 

12 

818681 

2-40 

876457 

1-84 

942223 

4 25 

057777 

067622 

48 

i3 

818825 

2-40 

876347 

1-84 

942478 

4-25 

4] 

i4 

818969 

819113 

2-40 

876236 

i-85 

942733 

4-25 

057267 

46 

i5 

2-40 

876125 

1 -85 

j 942988 

4-25 

067012 

45 

16 

819257 

2-40 

876014 

i-85 

! 943243 

4-25 

056757 

o565o2 

44 

»7 

819401 

2-40 

875904 

i-85 

943498 

4-25 

43 

10 

819545 

2-39 

875793 

i-85 

943762 

4-25 

056248 

42 

*9 

819689 

2-39 

875682 

i-85 

944007 

4-25 

055993 

055738 

4i 

20 

819832 

2-39 

875571 

i*85 

944262 

4-25 

4o 

21 

9.819976 

2-39 

9-875459 

1 *85 

9-944517 

4-25 

io-o55483 

3q 

22 

820120 

2-39 

875348 

i-85 

944771 

4-24 

055229 

38 

23 

820263 

2-39 

875237 

i-85 

945026 

4-24 

054974 

3] 

24 

820406 

2-39 

875126 

1 *86 

945281 

4-24 

054719 

36 

25 

82o55o 

2-38 

875014 

i-86 

945535 

4*24 

054465 

35 

26 

820693 

820806 

2-38 

874903 

i-86 

945790 

4-24 

054210 

34  1 

2-38 

874791 

i-86 

946045 

4-24 

o53955 

33  > 

26 

820979 

2-38  ! 

874680 

i-86 

946299 

4-24 

053701 

32 

29 

821122 

2-38  1 

874568 

i-86 

946564 

4-24 

053446 

3i 

3o 

821265 

2-38 

874456 

i-86 

946808 

4-24 

053192 

3o 

3i 

9-821407 

2-38 

9-874344 

i-86 

9-947063 

4-24 

10-062937 

29 

32 

,82i55o 

2-38 

874232 

1.87 

947318 

4-24 

052682 

28 

33 

821693 

2-37  I 

874121 

1.87 

947572 

4 • 24 

052428 

27 

34 

82i835 

2-37 

874009 

1-87 

947826 

4-24 

052174 

26 

35 

521977 

2-37  j 

873896 

1-87 

948081 

4-24 

051919 

25 

36 

822120 

2.37  | 

873784 

873672 

1-87 

948336 

4-24 

o5i664 

24 

37 

822262 

2.37  j 

1-87 

948590 

4-24 

o5i4io 

23 

38 

39 

822404 

2.37  1 

873560 

1-87 

948844 

4-24 

o5ii56 

22 

822546 

2.37  j 

873448 

1.87 

949099 

4-24 

050901 

21 

4o 

822688 

2-36 

873335 

1-87 

949353 

4*24 

050647 

20 

41 

9-822830 

2-36 

9-873223 

1-82 

9-949607 

4-24 

10 -050393 
o5oi38 

!9 

42 

822972 

2-36 

873110 

1 .88 

949862 

4-24 

l8 

43 

823114 

2-36 

872998 

1 .88 

9601 16 

4-24 

049884 

17 

44  I 

823255 

2-36 

872885 

i-88 

960870 

4-24 

049630 

10 

45  1 

823397 

2-36 

872772 

872669 

i-88 

950625 

4-24 

049375 

i5 

46 

823539 

2-36 

i-88 

950879 
951 i33  ! 

4*24 

049121 

14 

47  1 

823680 

2-35 

872547 

i-88 

4-24 

048867 

i3 

48 

823821 

2-35 

872434 

i-88 

95 1 388 

4-24 

0486 1 2 

12 

49 

823963 

2-35  ' 

872321 

i-88 

951642 

4-24 

048358 

11 

5o 

824*04 

2-35 

872208 

i-88 

961896 

4-24 

048104 

10 

\X 

9 824245 

2-35 

9-872095 

1-89 

9-952i5o 

4-24 

10-047860 

9 

5i 

824386 

2-35  j 

871981 

1 -89 

962405 

4-24 

047595 

8 

53 

54 

824527 

824668 

2 - 35 
2-34 

871868 

871755 

1-89 

1 -89 

962659 

962910 

4*24 

4-24 

047341 

047087 

046833 

2 

55 

824808 

2-34 

871641 

1-89 

953167 

4-23 

5 

56 

824949 

2-34 

871528 

1 -89 

953421 

4-23 

046579 

4 

825090 

82523c 

2-34 

871414 

1-89 

953676 

4-23 

046320 

3 

58 

! 2-34  ! 

871301 

1-89 

953929 

954180 

4-23 

046071 

2 

59 

825371 

I 2-34 

871187 

1-89 

4-23 

045817 

1 

60 

8a55i 1 

2-34 

871073 

1 -90 

954437 

4-23 

045564 

0 

Cosine 

L j>._ 

Si  no 

D. 

Cotang.  I 

D. 

Tang. 

M. 

(48  DEGRJSKO.) 


fiO  (42  DEGREES.;  A TABLE  OF  LOGARITHMIC 


M. 

0 

2 

3 

4 

5 

6 

I 

9 

IC 

11 

12 

13 

14 

15 

16 

\l 

19 

20 

21 

22 

23 

24 

25 

26 

ll 

II 

31 

32 

33 

34 

35 

36 

ll 

39 

40 

41 
4: 

43 

44 

45 

46 

s 

49 

50 

i 

52 

53 

54 

55 

56  | 

U\ 

& 

Sine 

D.  Coeine 

D.  | Tang. 

I). 

j Cotang. 

9*8255i 1 
82565i 
825701 
8259J1 
826071 
826211 
82635i 
826491 
826641 
826770 
826910 

9-827049 

827180 

827328 

827467 

827606 

827745 

827884 

828023 

828162 

828301 

9-828439 

828578 

828716 

828855 

828993 

829131 

829269 

829407 

829545 

829683 

9-829821 

829959 

830097 

830234 

83o372 

83o5o9 

83o64o 

830784 

830921 

83io58 

9-83i 196 
83i332 
831469 
83 1 606 
831742 

83 1 879 
832oi5 
832 1 52 
832288 
832425 

0-83256i 
832697 
832833 
832969 
833 io5 
833241 
833377 
833512 
833648 
833783 

2-34 

2-33 

2-33 

2-33 

2-33 

2-33 

2-33 

2-33 

2-33 

2-32 

2*32 

2-32 

2-32 

2-32 

2-32 

2-32 

2-32 

2-3l 

2-3i 

2-3l 

2-3i 

2-3l 

2 -3i 
2-3l 
2-3o 
2-3o 
2-3o 
2-3o 
2-3o 
2-3o 
2-3o 

2-29 

2-29 

2-29 

2-29 

2-29 

2-29 

2-29 

2-29 

2-28 

2-28 

2-28 

2-28 

2-28 

2-28 

2-28 

2-28 

2-27 

2-27 

2-27 

2-27 

2-27 

2-27 

2-27 

2-26 

2-26 

2-26 

2-26 

2-26 

2-26 

2-26 

9-871073 

870960 

870846 

870732 

870618 

870504 

870390 

870276 

870161 

870047 

869933 

9-869818 

869704 

869689 

869474 

869300 

869245 

869130 

869015 

868900 

868785 

9-868670 

868555 

868440 

868324 

868209 

868o93 

867978 

867862 

Sgg 

9-867515 

867399 

867283 

867167 

867051 

866935 

866819 

866703 

866586 

866470 

9-866353 

866237 

866120 

866004 

865887 

866770 

865653 

865536 

865419 

8653o2 

9-865i85 

865o68 

864950 

864§33 

864716 

864598 

864481 

864363 

864*45 

864127 

190 

1 -90 

1.90 

1 -90 

1 -90 

1 -90 

1 -90 

1 -90 

1 -90 
« -91 

1 *91 

1 *91 

191 

1.91 
191 

191 

1 1-91 

1. 91 

192 

1.92 
1-92 

1 -92 
1.92 
192 
1-92 
192 

1 -92 

1 -93 

,-93 

1 -93 

1 -93 

i-93 

1 -93 

1 -93 

1 -93 

1 -93 
1-94 
1-94 

1 -94 

1 -94 
1-94 

i-94 

i-94 

1-94 

1 -96 

1 -96 

1 -96 

1 -95 

1 -95 

1 -95 

1 -95 

1 -95 
i-95 

1 -95 

1 -96 

1 -96 

1 -96 
1-96 
1-96 

1 '96 
1-96 

1 9-954437 
95469 1 
954945 
955200 
955464 
955707 
i 966961 
956216 
956469 
966723 
956977 

9-957231 

957485 

95773o 

967993 

968246 

9585oo 

958754 

959008 

959516 

9 959760 
960023 

1 960277 

96o53i 
960784 
961038 
961291 
961545 
961709 
962062 

9-962306 

962560 

962813 

963067 

963320 

963574 

963827 

964081 

964335 

964588 

9-964842 

965095 

965349 

965602 

965855 

966105 

966362 

966616 
966869 
967 I 23 

9-967376 

967629 

96788J 

968136 

968389 

968643 

068896 

969140 

969403 

969656 

4*23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

1 4-23 

4-23 
4-23 
4-23 
4-23 
4-23 
4-23 
4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4 * 23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-23 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4*22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

4-22 

10 -045660 
o453oo 
o45o55 
044800 
044546 
o442q3 
044009 
043785 
04353 1 
043277 
043020 

10-042769 
o425i5 
042261 
042007 
041754 
04 1 5oo 
041246 
040992 
040708 

040484 

io-o4o23i 

039977 

o3g723 

039469 

o3o2i6 

038962 

038709 

o38455 

o382oi 

037948 

10-037694 
037440 
037187 
036933 
o3668o 
o36426 
o36n3  ! 
035919 
035665  | 
o354i2 

io-o35i58 
034905 
o3465 1 
034398 
o34i45 
033891 
o33638 
033384  I 
o33i3i 
032877  1 
10-032624 
032371 
032117 
o3i864 
o3 1 6 1 1 I 
o3i357 
o3iio4 
o3o85i 
o3o597 
o3o344 

60 

1 5? 

I 5?  1 
56 

55 

' 4 

53  1 
(?: 

% 

% 

45 

44 

43 

42 

41 

40 

II 

'll 

35 

I 34 

33 

1 33 

III 

ll 

II 

25 

24 

23 

22 

21 

20 

\l 

\l 

i5 

U 

i3 

12  | 
1 1 

10 

l 

l 

5 

4 

3 

2 

1 

0 

( ’-o*ino 

D. 

Sine  D.  I 

Cotang. 

I). 

Twig.  _ 1 M. 

(47  DKGKKKR.) 


SINES  AND  TANGENTS  (43  DEGREES.) 


61 


M. 

Sine 

D. 

Cosine 

1 D‘ 

Tang. 

1 D- 

1 Cotar  g. 

o 

9-833783 

2-26 

9-864^7 

1 -96 

9-969656 

4-22 

io-o3o344 

60 

i 

833919 

2-25 

864010 

1 06 

969909 

4-22 

030091 

5o 

2 

83*o54 

2-25 

863892 

1 '97 

970162 

4-22 

029838 

58 

3 

834 1 8q 

2-25 

863774 

1-97 

970416 

4-22 

029584 

57 

4 

834325 

2-25 

863656 

1-97 

970669 

4-22 

029331 

56 

83446o 

2-25 

863538 

1-97 

970922 

4-22 

029078 

55 

6 

834^5 

2-25 

863419 

1 *97 

971175 

4-22 

028826 

54 

7 

834740 

2-25 

8633oi 

! 1 *97 

971429 

4-22 

028571 

53 

8 

834865 

2-25 

863 1 83 

1.97 

971682 

4-22 

0283 1 8 

52 

9 

834999 

2 24 

863o64 

1.97 

971935 

4-22 

028065 

5i 

10 

835i 34 

2-24 

862946 

1 -98 

972188 

4-22 

027812 

5o 

ii 

9-835269 

2-24 

9-862827 

1 -98 

9-972441 

4-22 

10-027559 

49 

12 

8354o3 

2-24 

862709 

1 -98 

972694 

4-22 

027306 

48 

i3 

835538 

2-24 

862590 

1 -98 

97294^ 

4-22 

027052 

47 

14 

835672 

2-24 

862471 

1 -98 

973201 

4-22 

026799 

46 

i5 

835807 

2-24 

862353 

1 -98 

973454 

4-22 

026546 

45 

16 

835941 

2-24 

862234 

1 -98 

973707 

4-22 

026293 

44 

>7 

836o75 

2-23 

862ii5 

1 -98 

973960 

4-22 

026040 

43 

i8 

8362oo 

2-23 

861996 

1-98 

974213 

4-22 

025787 

42 

«9 

836343 

2-23 

861877 

1 -98 

974466 

4-22 

025534 

41 

20 

836477 

2-23 

861708 

1.99 

974719 

4-22 

j 025281 

40 

21 

9-8366i 1 

2-23 

9-86i638 

1-99 

9-974973 

4-22 

Iio-025o27 

39 

22 

836745 

2-23 

861629 

1 -99 

976226 

4-22 

024774 

38 

23 

836878 

2-23 

861400 

1-99 

976479 

4-22 

024521 

3] 

24 

837012 

2-22 

861280 

1 *99 

975732 

4-22 

024268 

36 

25 

837146 

2-22 

861 161 

1-99 

975985 

4-22 

024015 

35 

26 

837279 

2-22 

861041 

1-99 

976238 

4-22 

023762 

34 

27 

837412 

2-22 

860922 

1-99 

976491 

4-22 

m35o9 

33 

28 

837546 

2-22 

860802 

1-99 

976744 

4-22 

023256 

32 

19 

837679 

2-22 

860682 

2-00 

976997 

4-22 

o23oo3 

3i 

3o 

837812 

2-22 

86o562 

2-00 

977260 

4-22 

022750 

3o 

3i 

9 -557945 

2-22 

9 • 860442 

2-00 

9-9775o3 

4-22 

10  022497 

29 

32 

838078 

2-21 

86o322 

2-00 

977766 

4-22 

022244 

28 

33 

8382H 

2-21 

860202 

2-00 

978009 

4-22 

021991 

27 

34 

838344 

2-21 

860082 

2-00 

978262 

4-22 

021738 

26 

35 

838477 

2-21 

859062 

2-00 

9785 1 5 

4-22 

021486 

25 

36 

8386 10 

2-21 

859842 

2-00 

978768 

4-22 

021232 

24 

h 

838742 

2-21 

859721 

2-01 

979021 

4-22 

020979 

23 

38 

838875 

2-21 

859601 

2-01 

979274 

4-22 

020726 

22 

39 

839007 

2-21 

869480 

2-01 

979527 

4-22 

020473 

21 

4o 

839140 

2-20 

859360 

2-01 

979780 

4-22 

020220 

20 

4i 

9-839272 

2-20 

9-869239 

2-01 

9-980033 

4-22 

10-019967 

‘2 

42 

839404 

2-20 

859119 

2-01 

980286 

4-22 

019714 

l8 

43 

839536 

2-20 

868998 

2-01 

98o538 

4-22 

OI9462 

*7 

44 

839668 

2-20 

858877 

2-01 

980791 

4-21 

OI9209 

16 

45 

839800 

2-20 

858756 

2-02 

981044 

4-21 

OI8956 

i5 

46 

839932 

2-20 

858635 

2-02 

981297 

4-21 

018703 

14 

47 

840064 

2-19 

8585i4 

2-02 

981560 

4-21 

Ol845o 

i3 

48 

840196 

2-l9 

858393 

2-02 

981803 

4-21 

018197 

12 

49 

840328 

2-19 

868272 

2-02 

982056 

4-21 

017944 

1 1 

5o 

840459 

2-19 

858i5i 

2-02 

982309 

4-21 

OI769I 

10 

5i 

9-840591 

2-19 

9 -858o2o 

2-02 

9-982562 

4-21 

10  017438 

9 

52 

840722 

2-19 

857908 

2-02  ! 

982814 

4-21 

OIll86 

8 

53 

840854 

219 

857786 

2-02 

983067 

4-21 

010933 

7 

54 

840985 

2-19 

857665 

2 -o3 

983320 

4-21 

Ol668o 

6 

55 

841116  1 

2-18 

857543 

2-o3 

983573 

4-21 

016427 

5 

5 C 

841247 

2 • 18 

857422 

2-o3 

983826 

4-21 

016174 

4 

ll 

i 841378 

2-  l8 

8573oo 

2-o3 

984079 

4-21 

015921 

3 

56 

84 1 509 

2-l8 

857178 

2-o3 

984331 

4 * 21 

016660 

2 

J9 

841640 

2-18 

857056 

2-o3 

984684 

4-21 

016416 

1 

60 

84177* 

2 • 18 

856934 

2-o3 

1 

984837 

4-21 

0i5i63 

0 

! Cosine 

D. 

Sine 

D.  1 

Cotang. 

D. 

. Tang.  _ 

M. 

(46  DKOKERK.) 


62 


degrees.)  a table  of  logarithmic 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

1 

0 

9-841771 

a*  18 

9-856934 

2-o3 

9-984837 

4-21 

io-oi5i63 

60 

i 

841902 

a- 18 

8568ia 

2-o3 

986090 

4-21 

014910 

5o 

2 

842033 

218 

866690 

2 -04 

985343 

4-31 

014657 

58 

3 

842163 

2-17 

856568 

2-04 

9855 j6 

4-21 

014404 

57 

4 

842294 

2-17 

856446 

2 -04 

985848 

4-21 

0i4i5a 

% 

5 

842424 

2-17 

856323 

3- 04 

986101 

4-21 

013899 

55  ; 

6 

842555 

2-17 

856201 

2 -04 

986354 

4-21 

0 1 3646 

54  ! 

7 

842685 

217 

856078 

2- 04 

986607  | 

4-31 

013393 

1 53 

8 

842815 

2-  17 

855956 

2 o4 

986860 

4-21 

oi3i4o 

53 

9 

842946 

2-17 

855833 

2-04 

987112 

4-21 

012888 

5i 

10 

843076 

2-17 

855711 

2-o5 

987365 

4-21 

012635 

5o 

ii 

9-843206 

a- 16 

9-855588 

2-o5 

9-987618 

4-21 

10-012382 

4o 

ta 

843336 

a • 16 

855465 

2-o5 

987871 

4-21 

012129 

48 

i3 

843466 

a- 16 

855342 

2-o5 

988123 

4-21 

31 1877 

47 

14 

843595 

2-  16 

855219 

2-o5 

988376 

4-21 

01 1624 

46 

i5 

843725 

a- 16 

855o96 

2-o5 

988629 

4-21 

01 1371 

45 

16 

843855 

2 16 

854973 

2-o5 

988883 

4-21 

01 1 1 18 

44 

17 

843984 

2-  16 

854850 

2-o5 

989134 

4-21 

010866 

43 

18 

844114 

2 - 1 5 

854727 

2 06 

989387 

4-21 

oio6i3 

42 

*9 

844243 

2 1 5 

854^*3 

2- 06 

989640 

4- 21 

oio36o 

4i 

ao 

844372 

2-  l5 

85448o 

2- 06 

989893 

4-21 

010107 

40 

ai 

9 • 844502 

2 • I 5 

9-854356 

2- 06 

9-990145 

4-21 

10-009855 

39 

aa 

84463 1 

2 - I 5 

854233 

2-06 

990398 

990601 

4-21 

009602 

38 

23 

844760 

2 l5 

854109 

2- 06 

4-21 

009349 

37 

24 

25 

844889 

845oi8 

2 - I 5 

2-  l5 

853986 

853862 

2 06 
2- 06 

990903 
99 1 1 56 

4-31 

4-21 

009097 

008844 

36 

35 

26 

27 

845147 

845276 

2 • 1 5 

214 

853738 

853614 

2-  06 
2-07 

991409 

991662 

4-21 

4-21 

008591 

oo8338 

34 

33 

28 

8454o5 

214 

853490 

2-07 

991914 

4-21 

008086 

?2 

29 

845533 

2-14 

853366 

2-07 

992167 

4-21 

007833 

3i 

3o 

84566a 

2-14 

853242 

2-07 

992420 

4-31 

007580 

3o 

3i 

9-845790 

2-14 

9 • 853 1 18 

2-07 

9-992672 

4-21 

10-007328 

39 

3a 

845919 

214 

852994 

2-07 

992925 

4-21 

007075 

006822 

28 

33 

846047 

2-14 

852869 

2-07 

993178 

993460 

4-21 

27 

34 

846176 

214 

852745 

2-07 

4-21 

006570 

26 

35 

846304 

2-14 

852620 

2-07 

993683 

4-21 

006317 

25 

36 

846432 

2 • 1 3 

852496 

2- 08 

993936 

4-21 

006064 

24 

3t 

84656o 

2 • l3 

852371 

2 ■ 08 

994189 

4-21 

oo58i 1 

23 

38 

846688 

2-  l3 

852247 

2 • 08 

994441 

4-21 

oo5559 

22 

3g 

846816 

2-  l3 

852122 

2- 08 

994694 

4-21 

oo53o6 

21 

4o 

846944 

2 - 1 3 

851997 

2-08 

994947 

4-21 

oo5o53 

20 

4i 

9 847071 

2 • 1 3 

9.851873 

2- 08 

9 -995199 
996452 

4-31 

10-004801 

4a 

847199 

a- 13 

851747 

2 • 08 

4-21 

004548 

18 

43 

847327 

2 • 1 3 

85 1 622 

2- 08 

996705 

4-21 

004295 

>7 

44 

847454 

2 12 

85 1497 

2-09 

995957 

4-21 

004043 

16 

45 

84758a 

2-12 

85 1372 

2-09 

996210 

4-21 

003790 

003537 

oo3285 

i5 

46 

‘2 

847709 
847836 
847964 
K48091  j 

2-12 

312 

2-12 

85 1 246 

85ii2i 

2-09 

2 09 

996463 
9967 1 5 
996968 

4-21 

4-31 

14 

i3 

12 

oo3o3: 

48 

000090 

850870 

2 • 09 

4-21 

49 

212 

2-09  1 

997221 

4-21 

002779 

002627 

tl 

5o 

848218 

212 

850745 

2-09 

997473 

4-21 

10 

5i 

9 848345 

2-12 

9-850619 

2-09 

9 997726 

4-21 

10  00^274 

9 

5a 

84847a 

211 

860494 

210 

997979 

998231 

4-21 

002021 

8 

53 

848599 

2-11 

85o368 

2-10 

4-21 

001760 

ooi5i6 

7 

54 

848726 

2-11 

850243 

210 

998484 

4-21 

6 

55 

84S85a 

2-11 

85oi 16 

I 2 10 

998737 

4-21 

001263 

I 5 

56 

5} 

848970 

849106 

3-  II 

2-11 

849990 

849864 

210 

2-  10 

998989 

999242 

4-21 

4-21 

001011 

000758 

4 

3 

58 

84923a 

211 

849738 

2-10 

999495 

4-21 

000505 

2 

59 

849359 

849485 

2-11 

84961 1 

2-  10 

999748 

4-21 

000253 

1 

60 

2-11 

849485 

2-  10 

1 0 • 000000 

4-21 

10-000000 

0 

Cosine 

1 I). 

Sine 

1 I). 

| Cotang. 

L>.  | 

Tung. 

M. 

(46  DKORKKR.) 


A TABLE  OF  NATURAL  SINES. 


0 Deg. 

1 Deg. 

2 Deg. 

8 Deg. 

4 Deg. 

S. 

C.  S. 

S. 

C.  8. 

S. 

C.  8. 

S. 

C.  S. 

S. 

C.  8. 

M 

0 

00000 

Unit. 

01745 

99985 

03490 

99939 

o5234 

99863 

06976 

99756 

60 

I 

00029 

1 -0000 

01774 

99984 

03519 

99938 

o5263 

99861 

07005 

99754 

59 

1 

ooo58 

I *0000 

oi8o3 

99984 

o3548 

99937 

06292 

99860 

07034 

99752 

58 

3 

00087 

1 • 0000 

oi832 

99983 

03577 

99936 

o532i 

99868 

07063 

99750 

57 

4 

ooi  16 

1 • 0000 

01862 

9qq83 

o36o6 

99935 

o535o 

$5? 

07092 

99748 

56 

5 

ooi45 

1 -0000 

01891 

99982 

o3635 

99934 

05379 

07121 

99746 

55 

6 

00175 

I *0000 

01920 

99982 

o3664 

99933 

06408 

99854 

o-/i5o 

99744 

54 

7 

00204 

1 -OOOOi 

01940 

99981 

03693 

99932 

05437 

99852 

07170 

99742 

53 

8 

00233 

1 .0000 

01978 

qqq8o 

03723 

9993i 

o5466 

99851 

07208 

9974o 

52 

9 

00262 

I -0000 

02007 

09080 

03752 

99930 

o5495 

99849 

07237 

99738 

5i 

10 

00291 

1 .0000 

02036 

99979 

03781 

99929 

o5524 

99847 

07266 

99736 

5o 

ii 

00320 

99999 

02065 

99970 

o38io 

99927 

o5553 

99846 

07295 

99734 

49 

12 

oo34g 

99999 

02094 

99978 

o3839 

99926 

o5582 

99844 

07324 

9973i 

48 

i3 

00378 

99999 

02123 

99977 

o3868 

99926 

o56ii 

99842 

0-/353 

99729 

47 

14 

00407 

99999 

02  1 52 

99977 

99976 

03897 

99924 

o564o 

99841 

07382 

99727 

99725 

46 

i5 

oo436 

99999 

02181 

03926 

99923 

05669 

99839 

07411 

45 

16 

00465 

99999 

02211 

99976 

03955 

99922 

06698 

99838 

07440 

99723 

44 

•7 

00493 

99999 

02240 

99975 

03984 

99921 

05727 

99836 

07469 

99721 

43 

10 

oo524 

99999 

02269 

99974 

040 1 3 

99919 

06756 

99834 

07498 

99719 

42 

19 

oo553 

99998 

02298 

99974 

04042 

99918 

05-785 

99833 

07527 

99716 

4i 

20 

oo582 

99998 

02327 

99973 

04071 

999 1 7 

o58i4 

99831 

07556 

99714 

4o 

21 

0061 1 

99998 

02356 

99972 

04100 

99916 

06844 

99829 

07585 

99712 

3q 

22 

00640 

99998 

02385 

99972 

04129 

999 1 5 

05873 

99827 

99826 

07614 

99710 

38 

23 

00669 

99998 

02414 

99971 

041 5o 

999 1 3 

06902 

07643 

99708 

37 

24 

00698 

99998 

02443 

99970 

99969 

04188 

99912 

05931 

99824 

07672 

99705 

36 

25 

00727 

99997 

02472 

04217 

99911 

05960 

99822 

07701 

99703 

35 

26 

00756 

99997 

02501 

99960 

04246 

99910 

05980 

99821 

07730 

99701 

99699 

99696 

99694 

34 

37 

00785 

99997 

o253o 

99968 

04275 

99909 

06018 

99819 

°775o 

33 

20 

29 

00814 

00844 

99997 

99996 

o256o 

02589 

99967 

99966 

o43o4 

04333 

99907 

99906 

06047 

06076 

998 1 7 
99815 

07788 

32 

3i 

3o 

00873 

99996 

02618 

99966 

04362 

99906 

o6io5 

99813 

99692 

3o 

3i 

00902 

99996 

02647 

99965 

04391 

99904 

06 1 34 

99812 

07875 

99689 

29 

32 

00931 

99996 

02676 

99964 

04420 

99902 

06 1 63 

99810 

07904 

$3 

28 

33 

00960 

99995 

02705 

99963 

04449 

99901 

06192 

99808 

07933 

27 

34 

00989 

99995 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35 

01018 

99995 

02763 

99962 

o45o7 

04536 

99898 

o625o 

99804 

°1991 

99680 

25 

36 

01047 

99995 

02792 

99961 

99897 

99896 

06279 

o63o8 

99803 

08020 

99678 

24 

37 

01076 

99994 

02821 

99960 

04565 

9,9801 

08049 

08078 

99676 

23 

38 

oiio5 

99994 

0285o 

99969 

04594 

99894 

06337 

06366 

99799 

99673 

22 

39 

oii34 

99994 

028/9 

99959 

04623 

99893 

99797 

9.9790 

08107 

99671 

21 

40 

01 164 

99993 

02008 

99958 

04653 

99892 

06395 

o8i3o 

99668 

20 

41 

01193 

99993 

02938 

99967 

04682 

99890 

06424 

99793 

08 1 65 

99666 

19 

43 

01222 

99993 

02967 

99956 

04711 

Q9889 

99888 

06453 

99792 

08194 

99664 

18 

43 

OI25l 

99992 

02996, 

99955 

04740 

06482 

99700 

08223 

99661 

*7 

44 

01280 

99992 

o3o25 

99964 

04769 

99886 

o65ii 

99788 

08252 

99659 

16 

45 

01 309 

99991 

o3o54 

99953 

04798 

99885 

o654o 

99786 

08281 

99657 

i5 

46 

oi338 

99991 

o3o83  99952 

04827 

04856 

99883 

o65(>9 

99784 

o83io 

99654 

14 

47 

oi36t 

99991 

| 03lI2, 

99962 

99882 

06598 

99782 

08339 

99652 

i3 

48 

01396 

99990 

o3i4i 

9996i 

04885 

99881 

06627 

99780 

o8368 

99649 

12 

49  1 

01425 

99990 

03170' 

99950 

04914 

99870 

o6656 

99778 

08397 

99647 

11 

5o 

01454 

99989 

03199 

99949 

04943 

99878 

06685 

99776 

08426 

99644 

10 

ii 

oi483 

99989 

03228 

1 99948 

04972 

99876 

06714 

99774 

o8455 

99642 

9 

52 

oi5i3 

99980 

03257 

99947 

o5ooi 

99875 

06743 

99772 

08484 

99639 

8 

53 

54 

01 542 
01571 

99988 

99988 

03286 

o33i6 

1 99946 
99946 

o5o3o 

oSoSg 

99873 

99872 

06773 

06802 

99770 

99768 

o85i3 

08542 

2 

55 

01600 

99987 

o3345 

1 99944 

o5o88 

99870 

99869 

o683i 

99766 

08571 

99632 

5 

56 

01629 

oi658 

99987 

03374 

1 99943 

o5i  17 
o5i4o 

06860 

99764 

08600 

99630 

4 

*1 

99986 

o34o3 

99942 

9$l 

06889 

99762 

08629 

$3 

3 

58 

01687 

1 99986 

o3432 

99941 

o5i75 

06918 

99760 

o865§ 

2 

59 

01716 

, 99985 

o346i 

99940 

o52o5 

99864 

06947 

99758 

08687 

99622 

1 

M 

C.  8. 

hr” 

C.  S. 

S. 

C.  8. 

S. 

C.  S. 

S. 

C.  8. 

8 

M 

89  Deg. 

88  Deg. 

87  Deg. 

86  Deg. 

86  Deg. 

54 


A TABLE  OF  NATURAL  SINES. 


6 Deg. 

6 Deg. 

7 Deg. 

8 Deg. 

9 Deg. 

-1 

M 

8. 

| C.  8. 

8. 

c.  s. 

8. 

C.  8. 

8. 

C.  8 

S. 

G S. 

M 

0 

087161  99619 

io453 

994^2 

12187 

99255 

*3917 

99027 

15643 

98769 

60 

i 

08745 

99617 

10482 

99449 

12216 

i 9925 1 

13946 

99023 

15672 

98764 

59 

2 

08774 

99614 

io5i  1 

99446 

12245 

1 99248 

13975 

99019 

99016 

15701 

98760 

58 

3 

08803 

99612 

io54o 

99443 

12274 

99244 

14004 

1 5730 

98755 

57 

4 

o883i 

99609 

10569 

99440 

1 12302 

99240 

i4o33 

9901 1 

i5j58 

98751 

56 

5 

08860 

99607 

10597 

99437 

l i 233 1 

99237 

99233 

1 406 1 

99006 

98746 

55 

6 

08889 

99604 

1 0626 

99434 

1 I 2360 

14090 

99002 

98998 

98741 

54 

7 

08918 

99602 

io655 

9943 1 

i238o 

99230 

U119 

15845 

98737 

53 

8 

08947 

99599 

10684 

99428 

12418 

99226 

14148 

98994 

i5873 

15902 

98732 

52 

9 

08976 

99596 

10713 

99424 

12447 

99222 

*4177 

98990 

98986 

98728 

5i 

10 

09005 

99594 

10742 

99421 

12476 

99219 

i42o5 

15931 

98723 

5o 

1 1 

09034 

995qi 

1 077 1 

994i8 

i25o4 

99215 

14234 

98982 

15959 
1 5988 

987.8 

49 

12 

09063 

99588 

10800 

994i5 

12533 

9921 1 

14263 

98978 

987U 

48 

i3 

ogog2 

99586 

10829 

io858 

99412 

12562 

99208 

14292 

98973 

16017 

98709 

47 

i4 

09121 

99583 

99409 

99406 

12691 

99204 

14320 

16046 

98704 

46 

i5 

09150 

99580 

10887 

12620 

99200 

14349 

16074 

98700 

45 

16 

09170 

09208 

99578 

10916 

99402 

12649 

99197 

14378 

98961 

i6io3 

98695 

44 

*7 

99575 

10945 

99390 

99396 

12678 

99io3 

14407 

98957 

i6i32 

98690 

98686 

43 

18 

09237 

99572 

10973 

12706 

99189 

99186 

14436 

98953 

16160 

42 

19 

09266 

99570 

1 1002 

99393 

12735 

14464 

98948 

16189 

98681 

4i 

20 

09295 

99567 

iio3i 

$£ 

12764 

99182 

i4493 

98944 

16218 

98676 

4o 

21 

09324 

99564 

1 1060 

12793 

99178 

14522 

98940 

16246 

98671 

3q 

22 

09353 

99562 

11089 

99383 

12822 

99*7$ 

i455i 

98936 

16276 

98667 

38 

23 

09382 

99559 

1 1 1 1 8 

99380 

I285i 

99171 

99167 

i458o 

9893 1 

i63o4 

98662 

3? 

24 

09411 

99556 

11 147 

99377 

12680 

14608 

98927 

98923 

16333 

98657 

36 

25 

09440 

99553 

11176 

99374 

12908 

99163 

14637 

14666 

1 636 1 

98652 

35 

26 

09469 

9955i 

II2o5 

99370 

99367 

129J7 

99160 

98919 

16390 

98648 

34 

09498 

99548 

11234 

12966 

991 56 

14695 

98914 

16419 

98643 

33 

28 

09527 

09558 

99545 

11263 

99364 

12995 

99152 

14723 

98910 

16447 

98638 

32 

29 

99542 

1 1 291 

99360 

i3o24 

99148 

14752 

98906 

16476 

98633 

3i 

3o 

09585 

99540 

11320 

99357 

i3o53 

99*44 

14781 

98902 

i65o5 

98629 

3o 

3i 

09614 

99537 

11349 

11378 

99354 

i3o8i 

99141 

14810 

98897 

16533 

98624 

29 

32 

09642 

99534 

9935i 

i3iio 

99*37 

99133 

14838 

98893 

98889 

16562 

98619 

28 

33 

09671 

9953i 

11407 

99347 

i3i39 

14867 

16591 

98614 

27 

34 

09700 

99528 

11436 

99344 

1 3 168 

99129 

99125 

14896 

98884 

16620 

98609 

26 

35 

09720 

99526 

ii465 

99341 

13197 

13226 

14925 

98880 

16648 

98604 

25 

36 

09758 

99523 

11494 

99337 

99122 

14954 

98876 

16677 

16706 

98600 

24 

37 

2ZS 

99520 

1 1 523 

99334 

13254 

99118 

14982 

98871 

98867 

98863 

98695 

23 

38 

99517 

1 1 552 

9933i 

13283 

99**4 

i5oii 

16734 

98590 

98585 

22 

39 

09845 

995*4 

ii58o 

99327 

i33i2 

991 10 

i5o4o 

16763 

21 

4o 

09874 

995ii 

11609 
u 638 

99324 

i334i 

99106 

1 5069 

98858 

16792 

98580 

20 

4i 

09903 

99508 

99320 

13370 

99102 

S 

98854 

16820 

98575 

42 

09932 

99506 

11667 

11696 

99317 

13399 

99098 

98849 

16849 

98570 

l8 

43 

09961 

995o3 

Q93i4 

13427 

99094 

i5i55 

98845 

16878 

98565 

*7 

44 

09990 

99600 

11725 

99310 

13456 

9900* 

99087 

1 5i  84 

98841 

16906 

9856i 

16 

45 

10019 

99497 

11754 

99307 

13485 

1 52 1 2 

98836 

16935 

98556 

i5 

46 

10048 

99494 

11783 

993o3 

i35i4 

99083 

16241 

98832 

16964 

9855ii 

14 

4'1  1 
48  : 

10077 

10106 

99401 

99488 

11612 

11840 

99300 

99297 

13543 

13572 

99070 

99075 

15270 

15292 

98827 

98823 

16992: 
17021 1 

98546' 

q854i 

i3 

12 

49 

ioi35 

99485 

1 1869 

99293 

i36oo 

990]* 

99067 

99063 

99069 

99055 

15327 

98818 

I7o5o  985361 

n 

5o 

1 01 64 

99482 

11898 

%£ 

99283 

99279 

13629 

i5356 

98814 

170781 

9853i 

10  j 

51 

52 

53 

10192 

10221 

10250 

99479 

99476 

99473 

1 1985 

13658 

13687 

13716 

15385 

i54i4 

i5442 

98809 

98806 

98800 

171071 

17136 

17164 

98526 

98521 

98516 

i 

7 

54 

10279 

99470 

99467 

12014 

99276 

13744 

9905 1 

i547i 

98796 

17193 

98511 

0 

55 

io3o8 

12043 

99272 

99269 

i3773 

99047 

99043 

i55oo 

9879* 

17222 

98506 

5 

56 

10337 

io366 

99464 

12071 

13802 

15529 

98787 

17250 

98501 

4 

*7 

99461 

12 100 

99265 

1 383 1 

99039 

99035 

1 555-7 

98782 

17a  19! 

98496 

3 

58 

10395 

99458 

12129 

99262 

i386o 

1 5586 

98778 

17308 

98491 

98486 

2 

®9 

10424 

99455 

12158 

99258 

13889 

9903 1 

i56i5 

98773 

17336 

1 

V 

cTST 

8. 

C.  8. 

s7~ 

C.  8. 

8. 

C.  8. 

s. 

cTsTl 

S. 

M 

84  Deg. 

JMU^SL 

82  Deg. 

81  Deg 

80  Deg 

A TABLE  OF  NATURAL  SINES. 


66 


10  Deg. 

11  Deg. 

12  Dog. 

18  Deg. 

14  Deg. 

M 

S. 

C.  S. 

S. 

C.  S. 

S. 

C.  S. 

S. 

C.  S. 

S. 

C.  S. 

M 

0 

17365 

98481 

19081 

98163 

20791 

97815 

22495 

97437 

24192 

97o3o 

60 

i 

17393 

98476 

19109 

98157 

20820 

97800 

22523 

9743o 

24220 

97023 

a 

1742a 

98471 

19138 

98152 

20848 

97803 

22552 

97424 

24249 

97015 

58 

3 

i745i 

98466 

19167 

98146 

20877 

20905 

97797 

22580 

97417 

24277 

97008 

57 

4 

17479 

98461 

I9Iq5 

98140 

97791 

22608 

97411 

243o5 

97001 

56 

5 

17508 

98455 

19224 

98135 

20933 

97704 

22637 

97404 

24333 

96994 

96987 

55 

6 

\]lll 

98450 

19252 

98129 

20962 

97778 

22665 

97398 

24362 

54 

7 

98445 

19281 

98124 

20990 

97772 

22693 

97301 

24390 

96980 

53 

8 

17594 

98440 

19309 

98118 

21019 

97766 

22722 

97384 

24418 

96973 

5a 

9 

17623 

98435 

19338 

98112 

21047 

97760 

22760 

97378 

24446 

96966 

5i 

10 

17651 

98430 

ig366 

98107 

21076 

97754 

22778 

97371 

24474 

96969 

5o 

ii 

17680 

98425 

19395 

98101 

21104 

97748 

22807 

97365 

245o3 

96962 

49 

ia 

17708 

98420 

19423 

98096 

21 1 3 2 

97742 

22835 

97358 

2453i 

96945 

48 

i3 

17737 

98414 

19452 

98090 

21161 

97735 

22863 

9735i 

24559 

96937 

*7 

i4 

17766 

9840Q 

19481 

98084 

21189 

97720 

22892 

97345 

24587 

96930 

46 

45 

i5 

17794 

98404 

19509 

98079 

21218 

97723 

22920 

97338 

24615 

96923 

16 

17823 

q83oq 

19538 

98073 

21246 

97717 

22948 

9733i 

24644 

96916 

44 

17 

17852 

983(2 

19566 

98067 

21275 

9771 1 

22977 

97335 

24672 

96909 

43 

18 

17880 

98389 

98383 

19695 

98061 

2i3o3 

97706 

23oo5 

973i8 

24700 

96902 

42 

19 

17909 

19623 

98066 

2i33i 

97698 

23o33 

973u 

24728 

96894 

4i 

ao 

*7937 

98378 

19652 

98o5o 

2i36o 

97602 

23062 

973o4 

24756 

96887 

40 

ai 

17966 

98373 

19680 

98044 

21388 

97686 

23090 

97298 

24784 

96880 

3o 

aa 

98368 

I97°9 

98039 

21417 

97680 

23 1 1 8 

97291 

248i3 

96873 

38 

37 

a3 

98862 

19737 

98038 

21445 

97673 

23146 

97284 

24841 

96866 

a4 

i8o5a 

98357 

19766 

98027 

21474 

97667 

23175 

97278 

24869 

96858 

36 

-a5 

1 808 1 

98352 

19794 

98021 

2 1 5o2 

97661 

23203 

97271 

24897 

96851 

35 

a6 

18109 

98347 

19823 

98016 

2i53o 

97655 

23231 

97264 

24926 

96844 

34 

27 

18138 

98341 

19851 

98010 

21559 

97648 

23260 

97257 

24953 

96837 

33 

ao 

18166 

98336 

19880 

98004 

21687 

97642 

23288 

97251 

24982 

96829 

3a 

29 

18195 

9833. 

19908 

97998 

21616 

97636 

233 1 6 

97244 

25010 

96822 

3i ! 

3o 

18224 

98325 

19937 

97992 

21644 

97630 

23345 

97237 

25o38 

96816 

3o ! 

3i 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

97230 

i5o66 

96807 

20; 

3a 

18281 

983 1 5 

19994 

97981 

21701 

97617 

23401 

97223 

2 5094 

9680c 

28 

33 

18309 

983 10 

20022 

97975 

21729 

97611 

23429 

97217 

i 5 1 2 2 

96793 

96786 

27 1 

34 

1 8338 

983o4 

2oo5i 

97969 

21758 

97604 

23458 

97210 

a5i5i 

26 
25  I 

35 

18367 

98299 

20079 

97963 

21786 

97598 

23486 

97203 

25i79 

96778 

36 

18395 

98294 

20108 

97968 

218:4 

97592 

235i4 

97*96 

26207 

96771 

24! 

37 

18424 

98288 

2oi36 

97952 

21843 

97585 

23542 

97189 

25235 

, 96764 

23 

38 

18452 

98283 

aoi65 

97946 

21871 

97579 

23571 

97182 

25263 

96756 

22 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

21 

3o 

18609 

18538 

98272 

20222 

97934 

21928 

97566 

23627 

97169 

25320 

96742 

20 

4i 

98267 

20250 

97928 

21956 

97560 

23656 

97162 

25348 

96734 

42 

18567 

98261 

20279 

97922 

21985 

97553 

23684 

97i55 

25376 

96727 

18 

43 

l85g5 

98256 

2o3o7 

97916 

220l3 

97547 

23712 

97148 

26404 

96719 

*7 

44 

18624 

98250 

ao336 

97910 

22041 

97541 

23740 

97*4i 

26432 

96712 

16 

*5 

45 

18652 

98245 

2o364 

97906 

22070 

97534 

23769 

97*34 

25460 

96705 

46 

1 868 1 

98240 

20393 

97899 

97893 

22098 

97528 

23797 

97127 

25488 

96697 

14 

47 

18710 

98234 

20421 

22126 

97521 

23825 

97120 

255i6 

96690 

96682 

i3 

48 

1 28738 

i 98229 

2o45o 

97887 

221 55 

9751 5 

23853 

97**3 

25545 

i!  ; 

49 

I 18767 

! 98223 

20478 

97881 

22 1 83 

97508 

2388a 

97106 

25573 

96675 

11  , 

5o 

1 18795 

98218 

20607 

97875 

22212 

97502 

23gio 

97100 

256oi 

I 96667 

10, 

5i 

I 18824 

; 98212 

2o535 

22240 

97406 

97480 

97483 

97476 

23938,  97093 

256  29 

' 96660 

9| 

5a 

! i885a 

98207 

2o563 

22268 

23966 

97086 

25657 

1 96653 

8 

53 

54 

1 888 1 
18910 

| 98201 
, 98196 

20592 

20620 

97857 
9785 1 

22297 

22325 

239951  97079 
24023  97072 

25685 

25713 

96645 
9663  8 

55 

18938 

98190 

20649 

97845 

22353 

07470 

24o5i 

97065 

25741 

| 96630 

5 

56 

•8967 

98i85 

20677 

9783q 

97833 

22382 

97463 

24079 | 97058 
24108  9705 1 

25760 

96623 

4 

18995 

98179 

20706 

22410 

97457 

26798 

96615 

3I 

58 

19024 

! 98174 

20734 

97827 

22438 

9745o 

24 1 36.  97044 

25826 

96608 

a 

59 

19052 

98168 

20763 

1 97821 

22467 

97444 

24164 

97037 

25854 

1 96600 

1 

M 

C.  S. 

\~wr 

C.  S. 

1 8. 

c.  s. 

S. 

C.  S. 

i~s: 

C.  S. 

I~ST- 

M 

L 

7®  Deg. 

78  Deg. 

1 77  Deg. 

76  Deg. 

76  Deg. 

1 j 

A TABLE  OF  NATURAL  SINEB. 


% 


15 

Deg. 

16 

Deg. 

17 

Deg. 

18 

Deg. 

I 19  : 

Dog. 

M 

S. 

| C.  S. 

S. 

C.  S. 

S. 

| C.  8. 

“sT” 

"C.  8. 

8. 

I 8.C. 

M 

o 

35883 

965o3 

27564 

96126 

29237 

9563o 

30902 

96 1 dfe 

3a55-» 

9455a 

60 

i 

35910 

96585 

27592 

96118 

29265 

96622 

T0929 

95°97 

32584 

94542 

1 59 

l 

a5938 

1 96678 

27620 

96110 

29293 

956i3 

30957 

9508ft 

32612 

94533 

5ft; 

3 

a 5966 

; 96570 

27648 

96102 

29321 

956o5 

3o985 

96079 

II  32639 

1 94523 

5h 

4 

25994 

' 96562 

27676 

96094 

29348 

95596 

31012 

95070 

i 32667 

945 1 4 

561 

5 

a6oaa 

96555 

27704 

96086 

29376 

95588 

3io4o 

95061 

32694 

94604 

55' 

6 

a6o5o 

1 96547 

27731 

96078 

29404 

95579 

3io68 

96052 

1 32722 

94495 

54  | 

7 

36079 

96640 

27759 

96070 

1 29432 

g557i 

1 3lOg5 

95o43 

32749 

94485 

53, 

ft 

36107 

96532 

27787 

96062 

| 29460 

, 95562 

3 1 1 23 

95o33 

32777 

, 94476 

5a  1 

9 

a6i35 

96524 

27815 

96054 

29487 

95554 

3n5i 

96024 

32804 

94466 

5i  i 

10 

a6 1 63 

96517 

27843 

96046 

295.5 

95545 

1 31178 

96015 

32832 

I 94457 

5o 

1 1 

36191 

96609 

27871 

96087 

29543 

95536 

3 1 206 

96006 

32859 

94447 

49 

13 

36319 

96502 

27899 

96029 

2957i 

96628 

31233 

94997 

32887  94438 

48 

i3 

36247 

96494 

27927 

96021 

29J99 

1 955 19 

3i26i 

9498ft 

32914  94428 

47 

14 

36275 

96486 

27955 

96013 

29626 

g55 1 1 

31289 

94979 

32942 

94418 

46 

i5 

263o3 

96479 

27983 

96005 

29654 

955o2 

3 1 3 1 6 

94970 

32969 

94409 

45 

16 

2633i 

96471 

2801 1 

95997 

29682 

95493 

3 1 344 

94961 

32997 

94399 

44 

*7 

2635g 

96463 

28039 

95989 

29710 

95485 

3i372 

94962 

33o24 

94390 

43 

10 

26387 

96456 

28067 

96981 

29737 

95476 

3 1 3gg 

94943 

33o5i 

94380 

42 

*9 

264 1 5 

96448 

28095 

95972 

29766 

96467 

3l427 

94933 

33079 

94370 

41 

30 

26443 

96440 

28123 

95964 

29793 

96459 

31454 

94924 

33 1 06 

9436i 

40 

31 

26471 

96433 

28i5o 

95966 

29821 

9545o 

3l482 

94916 

33 1 34 

9435 1 

39 

33 

265oo 

96425 

28178 

95948 

29849 

95441 

3i5io 

94906 

33 161 

94342 

3ft 

33 

26528 

96417 

28206 

95940 

29876 

95433 

3 1 537 

94897 

33i89 

94332 

37 

24 

26556 

96410 

28234 

g5g3i 

29904 

95424 

3 1 565 

9488ft 

33210 

94322 

36 

35 

26584 

96402 

28262 

95923 

29932 

964 1 5 

3 1 5g3 

94878 

33244 

943 1 3 

35 

36 

26612 

96394 

28290 

95915 

29960 

95407 

31620 

94869 

33271 

943o3 

34 

27 

26640 

96386 

283 1 8 

95o°7 

29987 

95398 

31648 

94860 

33298 

94293 

33 

1 ™ 

26668 

96379 

28346 

95898 

3ooi5 

95389 

3 1 675 

9485i 

33326 

94284 

3a 

1 39 

26696 

96371 

28374 

95890 

30043 

9538o 

3 i7o3 

94842 

33353 

94274 

3i 

| 3o 

26724 

96363 

28402 

958ft2 

30071 

95372 

3 i73o 

94832 

3338i 

94264 

3o 

1 31 

26762 

96355 

28429 

96874 

30098 

95363 

3 1768 

94823 

334o8 

94254 

29 

33 

26780 

96347 

28457 

95865 

30126 

95354 

3i786 

94814 

33436 

94245 

28 

1 33 

26808 

96340 

28485 

95857 

3oi54 

95345 

3 1 81 3 

94805 

33463 

94235 

27 

I 34 

26836 

9633a 

285 1 3 

95849 

30182 

95337 

31841 

94795 

33490 

94225 

26 

35 

36864 

96324 

28541 

95841 

30209 

95328 

3 1868 

94786 

335 1 8 

942 1 5 

25 

36 

26892 

963i6 

28569 

95832 

30237 

95319 

3i8g6 

94777 

33545 

94206 

24 

37 

26920 

963o8 

28597 

95824 

30265 

953io 

3 1 923 

9476ft 

33573 

94i9^ 

23 

3ft 

26948 

96301 

28625 

958i6 

30292 

953oi 

3i95i 

94758 

336oo 

94186 

23 

39 

26976 

96293 

28652 

95807 

3o32o 

95293 

31979 

94749 

33627 

94176 

21 

4o 

27004 

96285 

28680 

95799 

3o348 

95284 

32006 

94740 

33655 

94167 

20 

4i 

27032 

96277 

28708 

9570! 

3o376 

95275 

32o34 

9473o 

33682 

94i57 

42 

27060 

96269 

28736 

95782 

3o4o3 

95266 

32061 

94721 

33710 

94147 

l8 

43 

27088 

96261 

28764 

95774 

3o43  1 

95267 

32089 

94712 

33737 

94i37 

17 

44 

27116 

96253 

28792 

95766 

3o45g 

95248 

32ii6 

94702 

33764 

94127 

l6 

45 

27144 

96246 

28820 

95757 

3o486 1 

95a4o 

32144 

94693 

33792 

9411ft 

i5 

46 

27172 

96238 

28847 

95749 

3o5i4 

96231 

32171 

94684 

33819 

94108 

14 

47  1 

27200 

96230 

28875 

95740 

3o542 

95222 

32199 

94674 

33846 

94098 

i3 

4ft 

37228 

96222 

28903 

95732 

3o57o 

952i3 

32227 

94665 

33874 

.-54088; 

12 

49 

27256 

96214 

28931 

95724 

3o597 

95204 

32254 

94656 

33901 

94078 

1 1 

5c 

27284 

96206 

28959 

95715 

3o625 

95196 

32282 

94646 

33929 

940681 

10 

5i 

27312 

96 1 98 

28987 

95707 

3o653 

95 1 86 

323o9 

94637 

33956| 

94o58‘ 

? 

5a 

27340 

96190 

29015 

95698 

3o68o 

95*77 

32337; 

94627 

33983 | 

94049: 

8 

53 

27368 

96182 

29042 

96690 

30708 

95 168 

323641 

94618 

34oi 1 , 

94039- 

7 

54 

27396 

96174 

29070 

9568 1 

3o736 

95 1 59 

32392  j 

94609 

34o38 

940291 

6 

55 

27424 

96166 

29098 

96673 

3o763 

95i5o 

32419 

94599 

34o65 

94019I 

5 

| 56 

27452 

961 58 

29126 

96664 

30791 

95142 

32447 

94590 

34093 

94009 

4 

*7 

27480 

96 1 5o 

29154' 

95656 

3o8ig 

g5 1 33 

32474 

94580 

34120 

93909 

3 

56 

27608 

96142 

29182 

95647 

30846 

95 1 24 

32502 

94571 

34147 

93989 

2 

59 

27536 

96134 

29309 

95639 

30874 

95 1 1 5 

3a529 

94561 

34170 

93979 

1 

M 

0.  S. 

8. 

C.  8.  1 8. 

C.  S. 

~sT~ 

G.  S. 

— s7— 

as. 

S. 

"M 

74  Dor. 

78  Dog. 

72  Deg. 

71  Deg. 

70  Deg. 

A TABLE  OF  NATURAL  SINES. 


20  Deg. 

21  Deg. 

22  Deg. 

28  Deg. 

24  Deg. 

M 

S. 

C.  S. 

S. 

C.  S. 

S. 

| C.  S. 

S. 

j C.  S. 

S. 

1 c,  s. 

0 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92o5o 

40674  9i355 

60 

1 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92039 

40700 

1 91343 

59 

3 

34267 

93949 

3589i 

93337 

37616 

92697 

39127 

Q2028 

40727!  9 1 33 1 

58 

3 

34284 

q3q3q 

35918 

93327 

37542 

92686 

39153 

92016 

40753  ai3i9 

57 

4 

343 1 1 

93929 

35945 

933i6 

37569 

9:675 

39i8o 

92005 

40780 

9*307 

56 

5 

34339 

Q3919 

35973 

933o6 

37595 

92  64 

39207 

| 9*994 

40806 

9*295 

55 

6 

34366 

q3qo9 

36ooo 

93295 

37022 

g2b53 

39234 

9 1 982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649  92642 

39260 

91971 

40860 

1 9*272 

53 

8 

34421 

98889 

36064 

93274 

37676 

9263l 

39287 

9^59 

40886 

91260 

1 5a 

9 

34448 

93879 

36o8i 

98264 

37703 

92620 

39314 

91948 

40913 

91248 

5i 

JC 

34475 

93869 

36io8 

93253 

37730 

92609 

39341 

91936 

40939 

91236 

5o 

ii 

345o3 

q3859 

36i  35 

93243 

37757 

925o8 

39367 

91925 

40966 

Q1224 

49 

13 

3453o 

98849 

36162 

93232 

37784 

92587 

39394 

91914 

40992 

91212 

48 

i3 

34557 

q3839 

36190 

93222 

3781 1 

92576 

39421 

91002 

41019 

91200 

47 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91891 

41045 

91188 

46 

i5 

34612 

938i9 

36244 

93201 

37865 

92554 

39474 

9*879 

41072 

91176 

45 

16 

34639 

93809 

36271 

93190 

37892 

92543 

39501 

91868 

41098 

91164 

44 

*7 

34666 

$$ 

36298 

93180 

37919 

92532 

39528 

9 1 856 

41125 

91 152 

43 

io 

34694 

36325 

93169 

37946 

92521 

39555 

91845 

41 1 5 1 

91 140 

42 

*9 

34721 

93779 

36352 

93 1 59 

37973 

925io 

39581 

9i833 

41178 

91128 

4i 

20 

34748 

93769 

36379 

93148 

3&g 

92400 

39608 

91822 

41204 

91 116 

4o 

21 

3477s 

93750 

36406 

93137 

92488 

39635 

91810 

41 23 1 

91104 

ll 

22 

348o3 

93748 

36434 

93127 

38o53 

92477 

39661 

91709 

41257 

91092 

23 

3483o 

93738 

3646i 

93116 

38o8o 

92466 

39688 

91787 

41284 

91080 

37 

24 

34857 

93728 

36488 

93106 

38107 

92455 

39715 

91775 

4i3io 

91068 

36 

25 

34884 

93718 

365 1 5 

93095 

38 1 34 

92444 

39741 

91764 

41337 

91066 

35 

26 

3491 2 

93708 

36542 

93084 

38i6i 

92432 

39768 

91762 

41 363 

91044 

34 

27 

34939 

93698 

36669 

93074 

38i88 

93421 

39795 

91741 

41390 

91032 

33 

28 

34966 

93688 

365g6 

93o63 

382 15 

92410 

39822 

9*720 

41416 

91020 

32 

29 

34993 

$8 

36623 

93o52 

38241 

92399 

39848 

91718 

41443 

91008 

3i 

3o 

35c2i 

3665o 

g3o42 

38268 

92380 

39875 

91706 

41469 

90996 

3o 

3i 

35o48 

93657 

36677 

93o3i 

38295 

92377 

39902 

91694 

4U96 

90984 

29 

32 

35o75 

93647 

36704 

93020 

38322 

92366 

39928 

91683 

4l522 

90972 

28 

33 

35io2 

93637 

36731 

93oio 

38349 

92355 

39955 

9*67* 

41549 

90960 

27 

34 

35i3o 

93626 

36758 

92999 

38376 

92343 

39982 

91660 

41575 

90948 

26 

35 

35 1 57 

93616 

36785 

92988 

3 84o3 

92332 

40008 

91648 

41602 

90936 

25 

36 

35 1 83 

93606 

368i2 

92978 

3843o 

92321 

4oo35 

9i636 

41628 

90924 

24 

37 

3521 1 

93596 

36839 

92967 

38456 

92310 

40062 

91625 

41 655 

90911 

23 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

91613 

41681 

90899 

22 

39 

35266 

93575 

36894 

92945 

385io 

92287 

4oii5 

91601 

41707 

90887 

21 

40 

36293 

93565 

36921 

92935 

38537 

92276 

4oi4i 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92266 

40168 

91578 

41760 

90863 

*9 

42 

35347 

93544 

36975 

92913 

38691 

92254 

40195 

91 566 

41787 

9085 1 

l8 

43 

35370 

93534 

37002 

92902 

38617 

92243 

40221 

9i555 

4i8i3 

90839 

*7 

44 

35402 

93524 

37029 

92892 

38644 

92231 

40248 

9i543 

41840 

90826 

l6 

45 

35429 

935i4 

37056 

92881 

38671 

92220 

40275 

9 1 53 1 

41866 

90814 

i5 

46 

35456 

935o3 

37083 

92870 

38698 

92200 

4o3oi 

91519 

41892 

90802 

14 

47 

35484 

934q3 

37110 

92859 

38725 

92198 

4o328 

9i5o8 

41919 

90790 

i3 

355 1 1 

93483 

37137 

92849 

38752 

92186 

4o355 

91496 

41945 

90778 

12 

49 

35538 

93472 

37164 

92838 

38778 

388o5 

92175 

4o38i 

9*484 

41972 

90766 

11 

5o 

35565 

93462 

37191 

92827 

92164 

40408 

91472 

41998 

90763 

10 

5i 

3559/ 

9345* 

37218 

92816 

38832 

92 1 52 

40434 

91461 

42024 

90741 

9 

53 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42o5i 

90729 

8 

53 

35647 

9343i 

37272 

92794 

92784 

92773 

38886 

92i3o 

40488 

9143] 

42077 

90717 

7 

54 

55 

35674 

35701 

93420 

I 934io 

37299 

37326 

38912 

38939 

921 19 
92107 

4o5i  4 
4o54i 

91426 

91414 

42104 

42i3o 

90704 

90692 

6 

5 

56 

3572(5 

; 93400 

37353 

! 92762 

38966 

92006 

92085 

4o5&7 

91402 

42 1 56 

90680 

4 

57 

35755!  93389 

37380 

j 92751 

38993 

40694 

91390 

42183 

90668 

3 

58 

35782 

37407 

| 92740 

39020 

92073 

40621 

91378 

42209 

90655 

2 

69 

35oio 

37434 

92729 

39046 

92062 

40647 

9i3o6 

42235 

90643 

1 

H 

C.  S. 

1 s. 

c.lT 

I S. 

CTsT 

S.  “ 

C.  S.' 

^sT~ 

"cTsT 

S. 

M 

l. 

69  Dqg. 

08  Deg. 

67  Deg 

66  Deg. 

65  Deg. 

19 


68 


A TABLE  OF  NATURAL  SINE8. 


25  Dog. 

26  Deg. 

27  Deg. 

28  Deg. 

29  Dog. 

M 

S 

c.  s. 

S. 

C.  S. 

S. 

C.  S. 

8. 

C.  8. 

S. 

I C.  8. 

M 

0 

42262 

9063 1 

43837 

89879 

45399 

89101 

46947 

88295 

484811  87462 

60 

i 

42288 

90618 

43863 

89867 

46426 

89087 

46973 

88281 

485o6 

, 87448 

1 59 

2 

423i5 

90606 

43889 

89854 

46461 

89074 

46999 

88267 

48532 

87434 

1 58 1 

3 

4234i 

90594 

43916 

89841 

46477 

46603 

89061 

47024 

88254 

48557 

, 87420 

57! 

4 

42367 

90582 

43942 

89828 

89048 

47060 

88240 

48583 

87406 

1 5o 

5 

42394 

90569 

43968 

89816 

45529 

89035 

47076 

88226 

48608 

87391 

55, 

6 

42420 

90557 

43994 

89803 

45554 

89021 

47101 

88213 

48634j  87377 
4865g  87363 

54 

7 

42446 

90545 

44020 

8979° 

46680 

89008 

47127 

88199 

88i85 

53 

8 

42473 

90532 

44046 

89777 

456o6 

88995 

47i53 

48684 

87349 

87335 

52 

9 

42499 

42526 

90520 

44072 

89764 

45632 

8898 1 J 

47178 

88172 

48710 

5i 

10 

90507 

44098 

89762 

45658 

88968! 

47204 

88 1 58 

48735 

87321 

5o 

1 1 

42552 

90496 

44124 

89739 

45684 

88955 ! 

47229 

88144 

48761 

87306 

49 

12 

42578 

90483 

441 5i 

89726 

46710 

88942 ] 

47255 

88i3oi 

48786 
4881 1 

87292 

48 

i3 

42604 

90470 

44177 

89713 

46736 

88928 

47281 

88117 

87278 

47 

i4 

4263 1 

90458 

44203 

89700 

45762 

8891 5; 

473o6 

88io3 

48837 

87264 

46 

i5 

42657 

90446 

44229 

89087 

46787 

88902 

47332 

88089 

48862 

87250 

, 45 

16 

4 2683 

90433 

44255 

89674 

458 1 3 

88888 

47358 

88075 

48888 

87235 

44 

•7 

42709 

90421 

44281 

89662 

46839 

46866 

88875 

47383 

88062 

48913 

87221 

43 

18 

42736 

90408 

44307 

89649 

88862! 

47409 

88048 

48938 

87207 

42 

19 

42762 

90396 

9o383 

44333 

89636 

46891 

88848 

47434 

88034 

48964 

87193 

4i 

20 

42788 

44359 

44385 

89623 

46917 

88835 

47460 

88020 

48989 

87178 

40 

21 

428i5 

90371 

89610 

46942 

88822 

47486 

88006 

49014 

87164 

3o 

22 

42841 

903  58 

4441 1 

89597 

45968 

88808 

475i  1 

87993 

49040 

87i5o 

38 

23 

42867 

90346 

44437 

89684 

46994 

88795 

88782 

47537 

; 87979 

49065 

87i36 

i 37 

24 

42894 

90334 

44464 

8967 ' 

46020 

47662 

87960 

49090 

87121 

1 36 

25 

42920 

903  2 1 

44490 

89558 

46046 

88768 

47688 

87951 

49116 

87io7 

j 35 

26 

42946 

90309 

44516 

89545 

46072 

88755 

47614 

87937 

49141 

87093 

34 

27 

42972 

90206 

90284 

44542 

89532 

46097 

88741 

47639 

87923 

49166 

87079 

33 

78 

42999 

43o25 

44568 

89619 

46123 

88728 

47665 

87909 

49  !92 

87064 

32 

29 

90271 

44594 

89506 

46149 

887i5 

47690 

87896 

49217 

87060 

3i 

3o 

43o5i 

90259 

44620 1 

89493 

46175 

88701 

47716 

87882 

49242 

87036 

3o 

3i 

43077 

90246 

44646' 

89480 

46201 

88688 

47741 

87868 

49268 

87021 

29 

32 

43io4 

90233 

44672 ! 

89467 

46226 

88674 

47767 

87854 

49293 

87007 

86993 

28 

33 

43i3o 

90221 

44698 1 

89454 

46252 

88661 

47793 j 
47818 

87840 

49318 

27 

34  1 

43 1 56 

90208 

44724, 

89441 

46278 

88647 

87826 

49344 

86978 

26 

35 

43i82 

90196 

4475oi 

89428 

46304 

88634 

47 844 ! 

87812 

49369 

86964 

25 

36 

43209 

90183 

44776 

894 1 5 

4633o 

88620 

47869 

87798 

49394 

86949 

86935 

24 

37 

43235 

901 7 1 

44802 

89402 

46355 

88607 

47895 

87784 

49419 

49445 

23 

38 

43261 

90i58 

44828 

89389 

4638i 

88593 

47920 

87770 

87756 

86921 

22 

39 

43287 

90146 

44864 

89376 

89363 

46407 

8858o 

47946 j 

49470 

86906 

21 

40 

433 1 3i 

90i33 

44880 

46433 

88566 

47971 

87743 

49496 

86892 

20 

41 

43340 

90120 

44906 

8935o 

46458 

88553 

47997 

48022 

87729 

49521 

86878 

*9 

42 

43366 

90108 

44932 

89337 

46484 

88539 

87715 

49546 

86863 

18 

43 

43392 

90095 

44968 

89324 

465io 

88526 

48048 

87701 

87687 

49671 

86849 

! 6 

! 44 

43418 

90082 

44984 

893 1 1 

46536 

885i2 

48073 

49596 

86834 

45 

43445 

90070 

45oio 

89298 

4656 1 

88499 

48099 

87678 

49622 

86820 

i5 

46 

43471 

90057 

90045 

45o36 

89285 

4658t 

88485 

48124 

87659 

49647, 

868o5i 

14 

47 

43497 

43523 

45o62 

89272 

89259 

4661 3 

88472 

481 5o 

87646 

49672 

86791 ! 

i3 

4« 

90032 

45o88; 

46639 

88458 

48175, 

87631 

49697! 

86777 

L2| 

49 

43549 

90019 

45i  14 

89245 

46664 

88445 

48201 

87617 

49723 

86762 

"I 

5o 

43575 

00007 

89994 

89981 
89968 
899  56 

45i4o 

89232 

46690 

88431 

48226 

87603 

49748 

86748 

0 

51 

52 

53 
34 

436o2 

43628 

43654 

4368o 

45 1 66 
46192 
45218 
45243 

89219 

89206 

89193 

89180 

46716 

46742 

46761 

46795 

88417 

88404 

88390 

88377 

88363 

48252 

£3 

48328 

87589 

87516 

37561 

87546 

49773! 

49798 

498241 

49849' 

867331 

86719 
86704 1 
86690 

i 

l 

55 

43706 

89943 

46269 

46296 

89167 

89i53 

46819 

48354 

87532 

49874 

86675 

5 

56 

43733 

89930 1 

46844 

88349 

48379 

48406 

875i8 

49899 

49924 

86661 

4 

57 

43769 

43785| 

89918 

45321 

89140 

46870 

88336 

87504 

866461 

3 

58 

89906 

89892 

45347 1 

89127 

46896 l 

88322 

48430 

87490 

49960 

86632 

> 

59 

438 11 

45373 

891 14 

4692 1 

883o8 

48456 

87476 

49975| 

86617 

* 

M 

c.  s.  "1 

S. 

cTsT 

S. 

c7  s. 

S. 

C.  8. 

S. 

C.  8.  I 

8. 

M 

04  Dog. 

I 08  Deg.  _ 

62  Dog. 

61  Deg. 

60  Deg. 

A TABLE  OF  NATURAL  SINES. 


69 


80  Deg. 

81  Deg. 

82  Deg. 

88  Deg. 

54  Deg. 

M 

S. 

C.  S. 

S. 

C.  S. 

S. 

I C.  S. 

S. 

C.  S 

S. 

C.  S. 

M 

0 

5oooo 

866o3 

5i5o4 

85717 

52992 

1 84805 

54464 

83867 

55910 

82904 

60 

i 

5oo25 

86588 

51629 

85702 

53oi7 

84789 

54488 

8385 1 

55943 

82887 

5* 

a 

5oo5o 

86573 

86559 

5 1 554 

85687 

53o4i 

84774 

84709 

84745 

545 1 3 

83835 

55968 

82871 

58 

3 

50076 

51679 

85672 

53o66 

54537 

838 19 

55992 

8a855 

57 

4 

5oioi 

86544 

5 1 604 

85657 

53091 

5456i 

838o4 

56oi6 

82839 

56 

' 5 

5oia6 

8653o 

51628 

85642 

53 1 1 5,  84728 

54586 

83788 

56o4c 

82822 

55 

6 

5oi5i 

865i  5 

5 1653 

85627 

53 140 

84712 

64610 

83772 

56o64 

82806 

54 

7 

50176 

865oi 

5i678 

85612 

53i64 

84097 

54635 

83756 

56o88 

82790 

53 

8 

50201 

86486 

51703 

86697 

53189 

84681 

54669 

54683 

8374o 

56i  12 

£# 

52 

9 

50227 

86471 

51728 

85582 

53214 

84666 

83724 

56i  36 

5i 

IO 

50252 

86407 

5i753 

85567 

53238 

84650 

54708 

837o8 

56 1 60 

82741 

5o 

11 

50277 

86442 

5 1 778 
5i8o3 

8555i 

53263 

84635 

54732 

83692 

56i84 

82724 

49 

u 

5o3o2 

86427 

85536 

53288 

84619 

54756 

83676 

8366o 

56208 

82708 

48 

i3 

5o327 

86413 

51828 

85521 

53312 

84604 

54781 

56232 

82692 

47 

14 

5o35a 

86398 

5i85a 

855o6 

53337 

84588 

548o5 

83645 

56256 

82675 

82669 

46 

i5 

5o377 

86384 

5i877 

85491 

5336i 

84573 

54829 

83629 

56280 

45 

16 

5o4o3 

86369 

51902 

85476 

53386 

84557 

54854 

836 1 3 

563o5 

82643 

44 

50428 

86354 

61921 

85461 

53411 

84542 

54878 

83597 

835oi 

56329 

56353 

82626 

43 

1 8 

5o453 

8634o 

61962 

85446 

53435 

84526 

54902 

82610 

42 

19 

50478 

86325 

51977 

8543i 

53460 

845 1 1 

54927 

83565 

56377 

82593 

41 

20 

5o5o3 

863 10 

52002 

85416 

53484 

84495 

5495 1 

83549 

56401 

82577 

40 

21 

5o528 

86295 

52026 

85401 

53609 

53534 

1 84400 

54975 

83533 

56425 

82561 

3o 

22 

5o553 

86281 

52o5i 

85385 

! 84464 

54999 

835i7 

56449 

82544 

38 

23 

5o578 

86266 

to 

O 

O 

85370 

53558 

84448 

55o24 

835oi 

56473 

82528 

37 

24 

5o6o3 

86261 

52101 

85355 

53583 

84433 

55o48 

83485 

56497 

8261 1 

36 

25 

50628 

86237 

52126 

8534o 

53607 

84417 

55o72 

83469 

83453 

56521 

82495 

35 

26 

5o654 

86222 

52 1 5 1 

85325 

53632 

84402 

55097 

56545 

82478 

34 

27 

60679 

86207 

62175 

853io 

53656 

84386 

55m 

83437 

56569 

82462 

33 

28 

50704 

86192 

52200 

85294 

5368 1 

84370 

55i45 

83421 

56593 

82446 

32 

29 

50729 

86178 

52225 

85279 

53705 

84355 

55169 

834o5 

56617 

82429 

82413 

3i 

1 3o 

50754 

86 1 63 

52250 

85264 

53*730 

84339 

55194 

83389 

56641 ^ 

3o 

| 3i 

50779 

86148 

52275 

85249 

53754 

84324 

55218 

83373 

56665* 

82396 

29 

1 32 

5o8o4 

86 1 33 

52299 

85234 

53779 

843o8 

55242 

83356 

56689 

82380 

28 

33 

50829 

86119 

52324 

85218 

538o4 

84292 

55266 

83340 

567 1 3 

82363 

27 

1 34 

5o854 

86104 

52349 

85203 

53828 

84277 

55291 

83324 

56736| 

82347 

26 

35 

50879 

86089 

62374 

i 85 1 88 

53853 

84261 

553 1 5 

833o8 

56760 

8233o 

25 

36 

50904 

86074 

62399 

1 85 i73 

53877 

84245 

55339 

55363 

83292 

56784 

568o8 

82314 

24 

37 

50929 

86o5o 

52423 

85i  67 

53902 

8423o 

83276 

82297 

23 

38 

50954 

86o45 

52448 

85i42 

53926 

84214 

55388 

83260 

5683 2 | 

82201 

22 

39 

5o979 

86o3o 

52473 

86127 

53961 

84198 

55412 

83244 

56856 

82264 

21 

40 

5 1 004 

860 1 5 

52498 

j 85 1 1 2 

53975 

84182 

55436 

83228 

5688o 

82248 

20 

41 

51029 

86000 

52522 

86096 

64000 

84167 

55460 

83212 

56904 : 

8223i 

*9 

42 

5io54 

85985 

52547 

85o8i 

54024 

! 841 5 1 

55484 

83 1 95 

56928 

82214 

IO 

! 43 

5 1 079 

85970 

85956 

62672 

85o66 

54o4o 

84 1 35 

55509 

83170 

66962 

82198 

n> 

44 

5iio4 

52697 

85o5i 

54073 

84120 

55533 

83 1 63 

56976 

82181 

16; 

45 

5i  1 29 

86941 

62621 

85o35 

54097 

84104 

55557 

83147 

57000] 

82165 

i5 

46 

5i  1 54 

85926 

52646 

85o2o 

54122 

84088 

5558i 

83i3i 

67024] 

81148 

14 

*7 

5i  179 

8691 1 

62671 

85oo5 

54146 

84072 

556o5 

83n5 

57047! 

82 1 32 

i3 

48 

5 1204 

85896 

52696 

84989 

54171 

84007 

5563o 

83098 

83o8a 

5707.1 

82115 

12 

49 

51229 

8588 1 

52720 

84974 

54195 

84041 

55654 

57095' 

82098 

11 

5o 

5i254 

85866 

52745 

84969 

8494J 

54220 

84025 

55678 

83o66 

id 

82082 

10 

5i 

5ij79 

8585 1 

52770 

54244 

84009 

55702 

83o5o 

82066 

9 

52 

5i3o4 

85836 

62794 

84928 

54269 

5429J 

83994 

55726 

83o34 

57167 

82048 

8 

53 

5 1 329 

85821 

62819 

84913 

83978 

83902 

5575o 

83017 

57191 

82032 

7 

54 

5i354 

858o6 

1 52844 

84897 

54317 

55775 

83ooi 

672 1 5 

82oi5 

6 

55 

56 

5i379 
5 1 404 

85792 
! 85777 

62869 

52893, 

84882 

84866 

54342 

54366 

83946 

83930 

lim 

82985 

82969 

82963 

57238 

57262 

81999 

81982 

5 

4 

57 

51429 

1 86762 

52918 

8485i 

54391 

83qi  5 

55847 

57286 

81965 

3 

58 

5i454 

q5747 

52943 

84836 

544i5 

83899 

83883 

5587i 

82936 

573 io| 

81949 

2 

*9 

5i479 

85732 

52967 | 

84820 

54440 

55895 

82920 

57334 

81932 

1 

M 

C.  S. 

P's- 

cTsTl 

“ sT“ 

C.  S. 

8. 

C.  S. 

~s7~| 

c.  s.  1 

s7“ 

M 

. 

| 69jDeg.  _ 

68  Deg. 

57  Deg. 

66  Deg. 

55  Deg. 

70 


A TABLE  OF  NATURAL  SINKS. 


13 

13 

14 

15 

16 

*9 

30 

31 
33 

33 

34 

35 

36 

11 

3i 

33 

33 

34 

35 

36 

u 

39 

! 4o 
4i 
43 
43 


85  Deg. 


07358 
5738i 
574o5 
5743o 
57453 

Vi11 

575oi i 
57534 
57548 

9 : A7?7? 
IC  57596j 

II  5-7619 
5-7643 
5766-7 1 
5-7691 
577 1 5j 
57738! 
57762! 
57o86 
578io| 
57833 
5785 7 
5788i 
57904 
57928 
57952 
57976 
57999 
58o23 
58o47 
58o7o 

58094 
081 18 
58i4i 
58i65i 
58i89! 
582121 
58236! 
68260! 
582831 
583o7I 
5833o 
58354 
58378 

44  ! 584oi 

45  j 58425 

46  ! 58449 

<7  58472 

4b  58496 
4c  1 585  m 
5c  58543 

5 1 58567 

52  ' 58590 

53  58614 
58637 
5866 1 
58684 
587o8 
5873i 
58755 
C.  8. 


C.  S. 

8mi5 

81899 

81882 

8 1 865 

81848 

8 1 832 

818 

81798 

81782 
8i765 
81748 
8 1 73 1 
81714 

81698 

81681 

81664 

81647 

8i63i 

81 614 

81697 
8i58o 
8 1 563 
8i546 
8i53o 
8i5i3 
81496 
81479 
81462 
8i445 
81428 
81412 

81396 
81 378 
8i36i 
81 344 
81 32  7 
8i3io 
81 293 
81276 
81269 
81242 
81225 
81208 
81191 

81 174 

8l  IDO 

8*  140 
3ii23 
81106 
81089 
81072 
8io55 
8io38 
81021 

81004 

80987 

80970 

80963 

80936 

80919 

“s7~ 


M Deg  .r>!i  Dog. 


S. 

58779 

58802 

58826 

58849 

58873 

58896 

58920 

58943 

58967 

68990 

69014 

69037 

59061 

59084 

59108 

69131 

69154 

59178 

59201 

59225 

69248 

59272 

59296 

593i8j 

59342 

69365! 

59389! 

594121 

59436 

59459 

59482 

69506 
69629 
59552 j 
695761 

59599 

59622 

59646 

59669 

59696 

59716 

59739 

59763 

5^! 

59832| 

59856l 
59879 ! 
59902, 
59926 
59949j 
,9972 

59995 

60019 
60042 
6006  5 
60089 
60112 
60 1 35 
60 1 58 


Deg. 

To.  s. 


c.  s. 


80902 

80885 

80867 

8o85o 

8o833 

80816 

80709 

80782 

3o765 

80748 

8o73o 

8o7i3 

80696 

80679 

8.652 

80644 

80627 

80610 

80693 

8o576 

8o558 

8o54 

8o524 

8o5o7 

80489 

80472 

8o455 

80438 

80420 

8o4o3 

8o386 

8o368 

8o35i 

8o334 

8o3i6 

80299 

80282 

80264 

80247 

8o23o 

80212 

80196 

80178 

80160 

8oi43 

8oi25 

80108 
80091 
8oo73 
8oo56 
8oo38 
80021 
8ooo3 
79986 
79968 
7995 1 

79934 

79016 
79809 
79881 
8.  ~ 


37  Deg. 

s.“  1 cTsT 


60182] 
60205! 
60228; 
6o25i ' 
60274] 
60298 
6o32i 
6o344 
6o36  7 
60390 
604 1 4 
60437 
60460 
60483 1 
6o5o6 


60629 

6o553 

6o576 

60699 

60622 

60645 

60668 

60691 

60714 

6o738 

60761 

60784 

60807 

6o83o 

6o853 

60876 

60899 

60922 

60945 

60968 

60991 

6ioi5 

6io38 

61061 

61084 

61 107 
61  i3o 
61 1 53 
61 176 
61199 
61222 

61245 

61268 

61291 

6 1 3 1 4 
61 337i 
6i36o 
61 383 
61406 
61429 
61 45 1 

61474 

61497 

6i52o 

61 543 

cTsT 


79864 
79846 
79829 
7981 1 
79793 
79776 
79758 
79741 
79723 
79706 
79688 
79671 
79653 
79635 
79618 
79600 

79583 
79565 
79547 
7953o 
795 1 2 
79494 
79477 
79459 
79441 
79424 
79406 
79388 
79371 
79353 
79335 
79318 
79300 
79282 
79264 
79247 
79229 
7921 1 
79193 
79176 
79 1 58 
79i4o 
79122 
79105 

79087 

79069 

7905 1 
79033 

7ooi5 

78998 

78980 

78962 

78944 

78926 

78908 

78891 

78873 

78855 

78837 

78819 

8. 


62  Deg. 


88  Deg. 

s7  c7s. 


61 566 
61689 
61612 
6i635 
6 1 658 
61681 
61704 
61726 

61749 

61772 

61795 

61818 

61841 

61864 

61887 

61909] 

61932 
6m55 
61978 
62001 
62024 
62046 
62069 
62092 
62 1 1 5 
62 1 38 
62160 
62i83 
62206 
62229 
6225i 

62274 

62297 

62320 

62342 

62365 

62388 

62411 

62433 

62456 

62479 

62502 

62524 

62547 

62670 

62592 

626i5 
62638 
62660 
62683] 
62706 
62728, 
6a75 1 1 
62774' 
62796 
62819 
62842 
62864 
62887 
62909 

c.  s. 


78801 

78783 

78765 

78747 

78729 

78711 

7869, 

78676 

78658 

78640 

78622 

78604 

78586 

78568 

7855o 

78532 

785i4 
78496 
78478 
78460 
78, 

78424 

784o5 
78387 
78369 
7835i 
78333 
783 1 5 
78297 
78279 
78261 

78243 
78225 
78206 
78188 
78170 
781 52 
78i34 
781 16 
78098 
78079 
78061 
78043 
78025 
78007 
77988 

77970 

77952 

77934 

77016 

77897 

77843 

77824 

77806 

77788 

77769 

7775i 

77733 

S. 


61  Dog. 


89  Deg.  | 

s.  Tend 

6-932]  777,5| 
62955]  77696, 
62977,  77678 
63 000,  77660] 
63o22  77641 
63o45  77623 
63o68  776o5 
630901  77586 
63 1 1 3 j 77568 
63 1 35  7755o 
63 1 58,  7753i 
63 180]  775i3 
632o3  77494 
63225]  77476 
63248 1 77458 
6327i|  77439 


63293 
633 1 6 
63338 
6336 1 
63383 
63406 
63428 
6345i 
63473 
63496 
635 1 8 
6354o 
63563 
63585 
636o8 

63o3o 
63653 
63675 
63698 
6372o 
63742 
63765| 
63787I 
638io] 
638321 
63854 
63877 
63899 
63922 
63944 

63966 
63989 
6401 1 
64o33 
64o56 
64078 
641001 
64 1 23 | 
64145 
64167I 
641901 

642  I 2| 

64234 

64256] 

C.  S.  I S. 

60  Deg. 


77421 

77402 

77384 

77366 

77347 

77329 

773io 

77292 

77IJ2 

772j5 

77236 

77218 

77199 

77181 

77162 

77144 

77125, 

77107 

77088 

77070 

77°5i 

77o33 

77014 

76996 

76977 

76969, 

76940 

769211 

76908! 

76884 

76866! 

76847 ' 

76828 

76810 

76791 

76773 

76754 

76735 

767*7 

76698 

76679 

76661 

76642 

76623 


5 

4. 

’! 

1 

M 


^ t.  u,wuiu>u,uiu<uia,y>^i  tv 

— OJ  0—1  OOO  O — *o  O— I OCsC  o — w>  <*>  O 0—1  oesO  O I 


A.  TABLE  OF  NATURAL  SINES. 


71 


40  Dog 

41 

Deg. 

42  : 

Deg. 

48 

Deg. 

44 

Deg. 

a 

S. 

C.  S. 

s. 

| C.  S. 

S. 

C.  S. 

S. 

| C.  S. 

S. 

1 a s. 

M 

0 

64279 

76604 

656o6 

75471 

66913 

74314 

6820c 

. 73 1 35 

69466 

, 71934 

60 

I 

643oi 

76586 

65628 

i 7545a 

66935 

74295 

68221 

73ii6 

69487 

1 71914 

5c 

2 

64323 

76567 

6565o 

75433 

66966 

74276 

68242 

73096 

69508 

71894 

3 

3 

64346 

76648 

65672 

1 75414 

66978 

74256 

68264 

1 73076 

69629 

: 71873 

57 

4 

64368 

76530 

66694 

l 75395 

66999 

74237 

68285 

73o56 

69549 

; 71853 

, 56 

5 

64390 

765ii 

65716 

i 75375 

67021 

74217 

683o6 

73o36 

1 69670 

l 71833 

55 

6 

64412 

76492 

65738 

75356 

67043 

74198 

68327 

73oi6 

69591 

I 71813 

; 54 

7 

64435 

76473 

66759 

75337 

67064 

74178 

68349 

72996 

69612 

i 71792 

53 

8 

64457 

76455 

65781 

753i8 

67086 

74i5g 

68370 

72976 

69633 

71772 

5a 

9 

64479 

76436 

658o3 

75209 

67107 

74139 

68391 

72957 

69654 

71752 

5i 

! 0 

64601 

76417 

65825 

75280 

67129 

74120 

68412 

72937 

69675 

71732 

5o 

II 

64524 

76398 

65847 

75261 

67151 

74100 

68433 

72017 

69696 

7171 1 

49 

ia 

64546 

76380 

65869 

76241 

67172 

74080 

68455 

72897 

69717 

71691 

48 

i3 

64568 

7636i 

65891 

75222 

67194 

74061 

68476 

72877 

69737 

71671 

47 

U 

64590 

76342 

65913 

752o3 

67216 

I 74o4i 

68497 

72857 

69758 

71660 

46 

1 5 

64612 

76323 

65935 

75184 

67237 

j 74022 

685 1 8 

72837 

69779 

7i63o 

45 

16 

64635 

76304 

65956 

75i65 

67258 

74002 

68539 

72817 

69800 

71610 

44 

*7 

64657 

76286 

65978 

75i46 

67280 

73983 

6856i 

72797 

69S21 

71690 

43 

18 

64679 

76267 

66000 

75126 

67301 

73963 

68582 

72777 

69842 

71669 

42 

*9 

64701 

76248 

66022 

75107 

67323 

73944 

686o3 

72757 

69862 

71549 

41 

ao 

64723 

76229 

66o44 

75o88 

67344 

73924 

68624 

72737 

69883 

71620 

40 

21 

64746 

76210 

66066 

76069 

67366 

73904 

68645 

72717 

69904 

7i5o8 

39 

aa 

64768 

76192 

66088 

"]5o5o 

67387 

73885 

68666 

72697 

69925 

71488 

38 

23 

64790 

76173 

66109 

75o3o 

67409 

| 73865 

68688 

72677 

69946 

71468 

3? 

24 

64812 

76164 

66i3i 

75oi  1 

67430 

73846 

68709 

72667 

69966 

71447 

36 

a5 

64834 

76i35 

66 1 53 

74992 

67452 

73826 

68730 

72637 

69987 

71427 

35 

26 

64856 

76116 

66175 

74973 

67473 

73806 

68751 

72617 

70008 

71407 

34 

27 

64878 

76097 

66197 

74953 

67495 

i3787 

68772 

72697 

70029 

71386 

33 

28 

64901 

76078 

66218 

74934 

67516 

73767 

68793 

72577 

70049 

7 1 366 

32 

29 

64923 

1 76069 

66240 

74915 

67538 

73747 

68814 

72557 

70070 

71345 

3i 

3o 

64945 

76041 

66262 

74896 

67559 

73728 

68835 

72537 

70091 

7 1325 

3o 1 

3! 

64967 

76022 

66284 

74876 

67580 

73708 

68857 

72617 

701 12 

7i3o5 

29 

3a 

64989 

76003 

663o6 

1 74857 

67602 

73688 

68878 

72497 

70132 

71284 

281 

33 

65oii 

75984 

66327 

74838 

67623 

73669 

68899 

72477 

7oi53 

71264 

27  i 

34  ! 

65o33. 

76965 

66349 

74818 

67645 

73649 

68920 

72467 

70174 

71243 

26 

35 

65o55 

75946 

66371 

74799 

67666 

73629 

68941 

72437 

70195 

71223 

25! 

36 

65077 

76927 

66393 

74700 

67688 

73610 

68962 

! 72417 

70215 

7 1 203' 

24! 

37 ! 

65099' 

76908 

66414 

74760 

67709 

73590 

68983 

72397 

70236 

71182 

23  1 

38 

65122 

73889 

66436 

74741 

67730 

735-70 

69004 

72377 

70267 

71162 

22  1 

39  ! 

65 1 44 

76870 

66458 

74722 

67752 

7355  j 

69025 

72357 

70271 

7ii4i 

21 

40  j 

65i66 

7585i 

: 66480 

74703 

67773 

7353 1 

69046 

72337 

70298 

71 1 2 1 

20 

41  i 

65 1 88 

75832 

I 665oi 

74683 

67195 

735ii 

69067  j 

72317 

70319 

71 100 

!9 

42 

652io 

758i3 

66523! 

74664 

67816] 

73491 

69088 

72297 

70339 

71080 

l8 

43 

65232 

75794 

66545 

74644 

67837 

73472 

69109 

72277 

7o36o 

71069 

17 

44 

65254 

75775 

66566 

74625 

67859! 

73452 

69130 

72257 

7o38i 

71039, 

10 

45 

65276 

76756J 

! 66588 

74606 

67880 

73432 

69i5i 

72236 

70401 

71019 

1 5 1 

46 

65298 

75738 

66610 

74686 

67901 1 

73412 

69172 

72216 

70422 

70998 

14 1 

<7  1 

65320 

76719 

66632 

74561 

67923 

73393; 

69193 

72196 

70443 

70978 

i3| 

65342 

76699 

66653 

74548 

67944' 

73373 

692141 

72176 

70463 

70957 

12 

49  ! 

65364 

75680 

66675 

74528 

67965 

73353 

69235] 

72 1 56 

70484 

70937 

11 ! 

5o  j 

65386 

75661 

66697 

74509 

67987 

73333 

69256 

721 35 

7o5o5 

70916 

wl 

, 5i  i 

654o8 

75642 

66718 

74489 

68008! 

733i4i 

69277! 

72116 

7o5a5 

70896 

91 

5a 

6543o 

75623 

66740 

74470 

68029* 

73294 

69298 

72096  1 

70646 

70875 

81 

53 

65452 

75604 

66762 

74451 

68o5i | 

732141 

69319! 

120]5  1 

70567! 

7o855 

/ 1 

54 

65474 

75585 

66783 

7443i 

68072: 

732541 

69340 

72005  i 

7°58i 

70834 

6; 

55 

66496 

75566 

668o5 

74412 

68093 

73234 

69361 

72035 

70608 

7081.3 

5 

56 

655 1 8 

76647 

66827 

74392 

681 1 5 

732 1 5, 

69382 

72015 

70628, 

70793 

4 

1 57 

65540 

75528 

66848 

743i3 

681 36 

73195 

69403 

7 if  96; 

70649 

70772, 

3! 

58 

6556a 

75509 

66870 

74353 

68i57 

73176 

69424 

71974I 

70670 

70762 

>! 

59 

65584 

75490 

66891 

74334 

68179 

73i55 

69445; 

71954 

70690 

7o73i 

1 

60 

656o6 

76471 

66913 

743i4 

68200! 

73 1 35 

69466] 

71934 

70711 

70711 

•j 

"m  ; 

as.  1 

" S. 

“cTsT 

S. 

~as.rl 

S.  ' 

a s7 1 

S. 

C.  S.  ! 

8. 

M 

1 

49  I>cg. 

43  Dog. 

47  Deg.  1 

Deg.  1 

_ 46  5*. 

I 

72 


TRAVERSE  table 


o 

r 

? 

i Deg. 

£ Deg. 

1 Deg. 

O 

5’ 

tO 

3 

1 

L&t. 

Dep 

Lat. 

Dep. 

Lat. 

Dep 

p 

3 

1 

1.00 

0.00 

1.30 

“OI 

'Too 

~0.01 

1 

2 

2.00 

0.01 

2.00 

0.02 

2.00 

0.03 

2 

3 

3.00 

0.01 

3.00 

0.03 

3.00 

0.04 

3 

4 

4.00 

0.02 

4.00 

0.03 

4.00 

0.05 

4 

6 

5.00 

0.02 

5.00 

0.04 

5.00 

0.07 

5 

6 

6.00 

0.03 

6.00 

0.05 

5.00 

0.08 

6 

7 

7.00 

0.03 

7.00 

0.06 

7.00 

0.09 

7 

8 

8.00 

0.03 

8.00 

0.07 

8.00 

0.10 

8 

9 

9.00 

0.04 

9.00 

0.08 

9.00 

0.12 

9 

10 

10.00 

0.04 

10.00 

0.09 

10.00 

0.  !3 

10 

11 

11.00 

0.05 

11.00 

0.10 

1 1.00 

0.14 

11 

12 

12.00 

0.05 

12.00 

0.10 

12.00 

0.16 

12 

13 

13.00 

0.06 

13.00 

0.11  . 

13.00 

0.17 

13 

14 

14.00 

0.06 

14.00 

0. 12 

14.00 

0.18 

14 

15 

15.00 

0.07 

15.00 

0.13 

15.00 

0.20 

15 

16 

16.00 

0.07 

16.00 

0.14 

16.00 

0.21 

16 

17 

17.00 

0.07 

17.00 

0.16 

17.00 

0.22 

17 

18 

18.00 

U 08 

18  00 

0.16 

18.00 

0.24 

18 

19 

19.00 

0.08 

19.00 

0.17 

19.00 

0.25 

19 

20 

20.00 

0.09 

20.00 

0.17 

20.00 

0.26 

20 

21 

21.00 

0.09 

21.00 

0.18 

21.00 

0.27 

21 

22 

22.00 

0.10 

22.00 

0.19 

22.00 

0.29 

22 

23  ] 

23.00 

0.10 

23.00 

0.20 

23.00 

0.30 

23 

24  1 

24.00 

0.10 

24.00 

0.21 

24.00 

0.31 

24 

25  j 

25.00 

0.11 

25.00 

0.22 

25.00 

0.33 

25 

26  ! 

26.00 

0.11 

26.00 

0.23 

26.00 

0.34 

26 

27 

27.00 

0.12 

27.00 

0.24 

! 27.00 

0.35 

27 

28 

28.00 

0.12 

28.00 

0.24 

28.00 

0.37 

28 

29 

29.00 

0.13 

29.00 

0.25 

29.00 

0.38 

29 

30 

30.00 

0.13 

30.00 

0.26 

30.00 

0.39 

30 

31 

31.00 

0.14 

31.00 

0.27 

31.00 

0.41 

31 

32 

32.00 

0.14 

32.00 

0.28 

32.00 

0.42 

32 

33 

33.00 

0.14 

33.00 

0-29 

33.00 

0.43 

33 

34 

34.00 

0.15 

34.00 

0.30 

34.00 

0.45 

34 

35 

35.00 

0.15 

35.00 

0.31 

35.00 

0.46 

35 

36 

36.00 

0.16 

36.00 

0.31 

36.00 

0.47 

36 

37 

37.00 

0.16 

37.00 

0.32 

37.00 

0.48 

37 

38 

38.00 

0.17 

38.00 

0.33 

38.00 

0.50 

38 

39 

39.00 

0.17 

39.00 

0.34 

! 39.00 

0.51 

1 39 

40 

40.00 

0.17 

40.00 

0.35 

| 40.00 

0.52 

40 

41 

41.00 

0.18 

41.00 

0.36 

41.00 

0.64 

41 

42 

42.00 

0.18 

42.00 

0.37 

42.00 

0.55 

42 

43 

43.00 

0.19 

43.00 

0.38  I 

43.00 

0.66 

i 43 

44 

44.00 

0.19 

44.00 

0.38  1 

44.00 

0.53 

44 

45 

45.00 

0.20 

45.00 

0.39 

45.00 

0.59 

45 

46 

46.00 

0.20 

46.00 

0.4C 

46.00 

0.60 

46 

47 

47.00 

0.21 

47.00 

0.41 

47.00 

0.62 

47 

48 

48.00 

0.21 

48.00 

0.42 

48.00 

0.63 

48 

49 

49.00 

0.21 

49.00 

0.43 

49.00 

0.64 

49 

50 

60.00 

0.22 

60.00 

0.44 

1 60.00 

0.65 

60 

8 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

CD 

O 

e 

§ 

Q 

89$  Deg. 

89|  Deg. 

89}  Deg. 

$ 

2 

Q 

TRAVERSE  TABLE 


73 


o 

oo’ 

p 

3 

O 

® 

i Deg. 

2 

Deg. 

1 1 Deg. 

| Distance.  | 

Lat. 

Dep. 

Lai. 

Dep. 

Lat. 

Dep. 

5] 

51.00 

1 0.22 

51.00 

~0~45 

706 

j ”6767 

51 

62 

52.00 

! 0.23 

52.00 

0.45 

52.00 

0.68 

52 

53 

53.00 

| 0.23 

53.00 

0.46 

| 53.00 

0.69 

i 53 

54 

54.00 

1 0.24 

54.00 

0.4  ; 

’ 54.00 

1 0.71 

54 

55 

55.00 

0.24 

55.00 

0.4o 

55.00 

! 0.72 

' 55 

56 

56.00 

0.24 

58.00 

0.49 

56.00 

0.73 

1 56 

5? 

57.00 

0.25 

57.00 

0.50 

57.00 

0.75 

1 57 

58 

58.00 

0.25 

58.00 

0.51 

57.99 

0.76 

58 

59 

59. 0C 

0.26 

59.00 

0.61 

' 58.99 

0.77 

59 

60 

60. 0( 

0.26 

60. 0C 

0.52 

, 59.99 

0.79 

60 

61 

61.00 

0.27 

61.00 

0.53“ 

60.99 

6.80 

61 

62 

62.00 

0.27 

62.00 

0.54 

61.99 

0.81 

62 

63 

63.00 

0.27 

63.00 

0.55 

62.99 

0.82 

63 

64 

64.00 

0.28 

64.00 

0.56 

63.99 

0.84 

64 

65 

65.00 

0.28 

65.00 

0.57 

64.99 

0.85 

65 

66 

66.00 

0.29 

66.00 

0.58 

65.99 

0.86 

i 66 

67 

67.00 

0.29 

67.00 

0.58 

66.99 

0.88 

67 

68 

68.00 

0.30 

68.00 

0.59  1 

67.99 

0.89 

68 

69 

69.00 

0.30 

69.00 

0.60 

68.99 

0.90 

69 

70 

70.00 

0.31 

70.00 

0.61 

69.99 

0.92 

70 

71 

71.00 

0.31 

71.00 

0.62 

70.99 

0.93 

•»i 

72 

72.00 

0.31 

72.00 

0.63 

71.99 

0.94 

72 

73 

73.00 

0.32 

73.00 ' 

0.64 

72.99 

0.96 

73 

74 

74.00 

0.32 

74.00 

0.65 

73.99 

0.97 

74 

75 

75.00 

0.33 

75.00 

0.65 

74.99 

0.98 

75 

76 

76.00 

0.33 

76.00 

0.66 

75.99 

0.99 

76 

77 

77.00 

0.34 

77.00 

0.67 

1 76.99 

1.01 

77 

78 

78.00 

0.34 

78.00 

0.68 

i 77.99 

1.02 

78 

79 

79.00 

0.34 

79.00 

0.69 

1 78.99 

1.03 

79 

80 

80.00 

0.35 

80.00 

0.70 

1 79.99 

1.05 

80 

81 

81.00 

0.35 

81.00 

0.71 

80.99 

1.06 

'81 

82 

82.00 

0.36 

82.00 

0.72 

81.99 

1.07 

82 

83 

83.00 

0.36 

83.00 

0.72 

82.99 

1.09 

83 

84 

84.00 

0.37 

84.00 

0.73 

83.99 

1.10 

84 

85 

85.00 

0.37 

85.00 

0.74 

84.99 

1.11 

85 

86 

86.00 

0.38 

86.00 

0.75 

85.99 

1.13 

86 

87 

87.00 

0.38 

87.00 

0.76 

86.99 

1.14 

87 

88 

88.00 

0.38 

88.00 

0.77 

87.99 

1.15 

88 

89 

89.00 

0.39 

89.00 

0.78 

88.99 

1.16  . 

89 

90 

90.00 

0.39 

90.00 

0.79 

89.99 

1.18  1 

90 

91 

91.00 

6.40 

91.00 

0.79 

90.99 

1.19  ' 

91 

92 

92.00 

0.40 

92.00 

0.80 

91.99 

1.20  | 

92 

93 

93.00 

0.41 

93.00 

0.81 

92  99 

1.22  ! 

93 

94 

94.00 

0.41 

94.00 

0.82 

93  99 

1 23  1 

94 

95 

95.00 

0.41 

95.00 

0.83 

94.99 

1.24  , 

95 

96 

96.00 

0.42 

96.00 

0.84 

95.99 

1.26 

96 

9? 

97.00 

0.42 

97.00 

0.85  ! 

96.99 

1.27 

97 

98 

98.00 

0.43 

98.00 

0.86 

97.99 

1.28 

98 

99 

99.00 

0.43 

99.00 

0.86 

98.99 

1.30 

99 

'00 

100.00 

0.44 

100.00  | 

0.87 

99.99 

1.31 

.00 

© 

e 

a 

aj 

«B 

Q 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

$ 

a 

i 

89  ] Deg. 

89£  Deg. 

89|  Deg. 

74 


TItAV I'.KSE  TABLE 


g 

5 

r-* 

1 Deg. 

H Deg. 

1*  Dog. 

U 

Deg. 

j 

1 

S 

o 

Lat. 

Dep. 

Lat.  | Defp. 

Lat. 

Dep 

Lat. 

Dep.  j 

f 

1 

1.00 

0.02 

1.00 

0.02 

1.00 

0 03 

1 .00  1 

0703* 

2 

2.00 

0.03 

2.00 

0.04 

2.00 

r,jn 

2.00 

0.06 

2 

3 

3.00 

0.05 

3.00 

0.07 

3 00 

O.Ow 

3.00 

0 09  , 

3 

4 

4.00 

0.07 

4.00 

0.09 

1 00 

0.10 

4.00 

0.12 

1 

6 

5.00 

0.09 

5.00 

0.  1 l 

1 5.00> 

0.13 

5.00  | 

0.15 

5 

0 

6.00 

0.10 

6.00 

0. 13 

6. 00 

0.16 

6.00 

0 18 

6 

7 

7.00 

0.12 

7.00 

0.  15 

7.00 

0.18 

7.00 

0.21 

7 

8 

8.00 

0.14 

8.00 

0.  17 

8.00 

0.21 

8.00 

0.25 

8 

9 

9.00 

0.16 

9.00 

0.20 

9.00 

0.24 

9.00 

0.28 

9 

10 

10.00 

0 17 

10.00 

0.22 

10.00 

0.26 

10.00 

0.31 

10 

11 

11.00 

0.19 

J 1.00 

0.24 

11.00 

0.28 

10.99 

0.34 

11 

12 

12.00 

0.21 

12.00 

0.26 

12.00 

0.31 

11.99 

0.37 

12 

13 

13.00 

0.23 

13.00 

0.28 

13.00 

0.34 

12.99 

0.40 

13 

14 

14.00 

0.24 

14.00 

0.31 

14.00 

0.37 

13.99 

0.43 

14 

15 

15.00 

0.26 

15.00 

0.33 

14.99 

0.39 

14.99 

0.46 

i5 

16 

16.00 

0.28 

16.00 

0.35 

15.99 

0.42 

15.99 

0.49 

16 

17 

17.00 

0.30 

17.00 

0.37 

16.99 

0.45 

16.99 

0.52 

17 

18 

18.00 

0.31 

18.00 

0.39 

17.99 

0.47 

17.99 

0.55 

18 

19 

19.00 

0.33 

19.00 

0.41 

18.99 

0.50 

18.99 

0.58 

19 

20 

20.00 

0.35 

20.00 

0.44 

19.99 

0.52 

19.99 

0.61 

20 

21 

21.00 

0.37 

21.00 

0.46 

20.99' 

0.55 

20.99 

0.64 

21 

22 

22.00 

0.38 

21.99 

0.48 

21.99 

0.58 

21.99  1 

0.67 

22 

23 

23.00 

0.40 

22.99 

0.50 

22.99 

0.60 

22.99 

0.70 

23 

24 

24.00 

0.42 

23.99 

0.52 

23.99 

0.63 

23.99 

0.73 

24 

25 

25.00 

0.44 

24.99 

0.55 

24.99 

0.65 

24.99 

0.76 

25 

26 

26.00 

0.45 

25.99 

0.57 

25.99 

0.68 

1 25.99 

0.79 

26 

27 

27.00 

0.47 

26.99 

0 59 

26.99 

0.71 

1 26.99 

0.83 

27 

28 

28.00 

0.49 

27.99 

0.61 

27.99 

0.73 

1 27.99 

0.86 

28 

29 

29.00 

0.51 

28.99 

0.63 

28.99 

0.76 

28.99 

0.89 

29 

30 

30.00 

0.52 

29.99 

0.65 

29.99 

0.79 

29.99 

0.92 

30 

31 

31.00 

0.54 

30.99 

0.68 

30.99 

0.81 

30.99 

0.95 

31 

32 

32.00 

0.56 

31.99 

0.70 

31.99 

0.84 

31.99 

0.98 

32 

33 

32.99 

0.58 

32.99 

0.72 

32.99 

0.86 

32.98 

1 1.01 

33 

34 

33.99 

0.59 

33.99 

0.74 

33.99 

0.89 

33.98 

1.04 

34 

35 

34.99 

0.61 

34.99 

0.76 

34.99 

0.92 

34.98 

1.07 

35 

36 

35.99 

0.63 

35.99 

0.79 

35.99 

0.94 

35.98 

1.10 

36 

37 

36.99 

0.65 

36.99 

0.81 

36.99 

0.97 

36.98 

1.13 

37 

38 

37.99 

0.66 

37.99 

0.83 

37.99 

0.99 

37..  98 

1.16 

38 

39 

38.99 

0.68 

38.99 

0.85 

38.99 

1.02 

38.98 

1.19 

39 

40 

39.99 

0.70 

39.99 

0.87 

39.99 

1.05 

39.98 

! 1.22 

40 

1 41 

40.99 

0.72 

40.99 

0.89 

40.99 

1.07“ 

40.98 

1 25 

'41 

1 42 

41.99 

0.73 

41.99 

0.92 

41.99 

1.10 

41.98 

1 28 

42 

43 

42.99 

0.75 

42.99 

0.94 

42.99 

1.13 

42.98 

1.31 

43 

44 

43.99 

0.77 

43.99 

0.96 

43.99 

1.15 

43.98 

1.34 

14 

45 

1 44.99 

! 0.79 

44.99 

0.98 

44.99 

1.18 

44.98 

1.37 

45 

46 

45.99 

0.80 

45.99 

1.00 

45 . 99 

1.20 

45.98 

1.40 

46 

47 

1 46.99 

0.82 

46.99 

1.03 

46.99 

1.23 

46.98 

1.44 

47 

46 

47.99 

0.84 

47.99 

1.05 

47.98 

1.26 

47.98 

1 47 

48 

41 

. 48.99 

0.86 

48.99 

1.07 

48.98 

1.28 

48.98 

1.50 

49 

50 

49.99 

0.87 

49.99 

1.09 

49.98 

1.31 

1 49.98 

1.53 

50 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

© 

o 

c 

1 

Q 

09  Deg. 

88;}  Deg. 

88| 

Deg. 

881  Deg. 

1 

a 

■n 

5 

TRAVERSE  TABLE 


75 


Distance. 

1 Deg.  1 

Li 

H Deg. 

4 Deg. 

1 Deg. 

Distance. 1 

Lat. 

Dep. 

It. 

Dep. 

Lat. 

j Dep. 

Lat. 

Dep. 

51 

50 

99 

0 

89 

50 

99 

1.11 

50.98 

1.34 

50. 

,98 

1.66 

61 

52 

51 

99 

0 

91 

51 

99 

1.13 

51.98  j 

I 1.36 

51. 

,98 

1.59 

62 

63 

52 

99 

0 

92 

52 

99 

1.16 

52.98 

1.39 

52. 

98 

1.62 

53 

54 

53 

99 

0 

94 

53 

99 

1.18 

53.98 

1.41 

53. 

97 

1 . 65 

51 

55 

54 

99 

0 

96 

54 

99 

1.20 

54.98 

1.44 

54. 

97 

1.68 

56 

56 

55 

99 

0 

98 

55 

99 

1.22 

55 . 98 

1.47 

55. 

97 

1.71 

56 

5? 

56 

99 

0 

99 

56 

99 

1.24 

56  98 

1.49 

56. 

97 

1.74  | 

67 

58 

57 

99 

] 

01 

57 

99 

1.27 

57 . 98 

1.52 

57. 

97 

..77 

58 

59 

58 

99 

1 

03 

58 

99 

1.29 

58.98 

1.54 

58. 

97 

1.80 

69 

60  ! 59 

99 

1 

05 

59 

99 

1 .31 

59.98 

1.57 

59. 

97 

1.83 

60 

61 

60 

99 

1 

06 

60 

99 

1.33 

60.98 

1 .60 

60. 

97 

" 1 .86 

61 

62 

61 

99 

1 

08 

61 

99 

1.35 

61.93 

1.62 

61. 

97 

1.89 

62 

63 

62 

99 

1 

10 

62 

99 

1.37 

62.98 

1.65 

62. 

97 

1.92  1 

63 

64 

63 

99 

1 

12 

63 

98 

1.40 

63.93 

1.68 

63. 

97 

1.95 

64 

65 

64 

99 

1 

13 

64 

98 

1 .42 

64.98 

1.70 

64. 

97 

1 89 

65 

66 

65 

99 

l 

15 

65 

98 

1.44 

85.98 

1.73 

65. 

97 

2 02  , 

66 

67 

66 

99 

1 

17 

66 

98 

1.46 

66.98 

1.75 

66. 

97 

2.05  j 

67 

68 

67 

99 

1 

19 

67 

98 

1.48 

67.98 

1.78 

67. 

97 

2.08  j 

68 

69 

68 

99 

1 

20 

68 

98 

1.51 

68.98 

1.81 

68 

97 

2.11  ! 

69 

70 

69 

99 

1 

22 

69 

98 

1.53 

69.98 

1.83 

69. 

97 

2.14 

70 

71 

70 

99 

1 

24 

70 

98 

1.55 

70.98 

1.86 

70. 

97 

2.17 

71 

72 

71 

99 

1 

26 

71 

98 

1.57 

71.98 

1.88 

71. 

,97 

2.20 

72 

73 

72. 

.99 

1, 

.27 

72. 

,98 

1 .59 

72.97 

1.91 

72. 

97 

2.23 

73 

74 

73. 

.99 

1, 

.29 

73. 

.98 

1 .61 

73.97 

1.94 

73. 

,97 

2.26 

74 

75 

74, 

, 99 

1 

.31 

74, 

.98 

1.64 

74.97 

1.96 

74. 

,97 

2.29 

75 

76 

75, 

,99 

1, 

.33 

75, 

.98 

1.66 

75.97 

1.99 

75. 

,96 

2.32 

76 

77 

76, 

.99 

1, 

.34 

76 

.98 

1 .68 

76.97 

2.02 

76. 

,96 

2.35 

77 

78 

77, 

.99 

1 , 

.36 

77, 

.98 

1.70 

77.97 

2.04 

77. 

,96 

2.38 

78 

79 

78, 

.99 

1 , 

.38 

78, 

.98 

1.72 

78.97 

2.07 

78, 

.96 

2.41 

79 

80 

79, 

.99 

1 

.40 

79 

.98 

1.75 

79.97 

2,09 

79, 

.96 

2.44 

80 

81 

80, 

,99 

] 

.41 

so , 

.98 

1.77 

80.97 

2.12 

80, 

.96 

2.47 

81 

82 

81, 

.99 

1 , 

.43 

81 , 

.98 

1.79 

81.97 

2.15 

81, 

.96 

2.50 

82 

83 

82, 

,99 

1, 

.45 

82 

.98 

1 .81 

82.97 

2.17 

82, 

.96 

2.53 

83 

84 

83, 

.99 

1 

.47 

83, 

.98 

1.83 

83.97 

2.20 

83, 

.96 

i 2.57 

84 

85 

84, 

.99 

1 

.48 

84 

.98 

1 .85 

84.97 

2.23 

84 

.96 

2.60 

85 

86 

85, 

.99 

1 

.50 

85 

.98 

1 .88 

85.97 

2.25 

85 

.96 

2.63 

86 

87 

86 

.99 

1 

.52 

86 

.98 

1.90 

86.97 

2.28 

86 

.96 

2.66 

87 

88 

87, 

.99 

1 

.54 

87 

.98 

1.92 

87.97 

2.30 

87 

.96 

2.69 

88 

89 

88 

.99  1 

1 

.55 

88 

.98 

1.94 

88.97 

2.33 

88 

.96 

2.72 

89 

90 

i 89 

.99  1 

1 

.57 

89 

.98 

1.96 

89.97 

| 2.36 

89 

.96 

2.75 

90 

91 

i 90 

.99  ! 

1 

.59 

90 

.98 

1 .99 

90.97 

! 2.38 

90 

.96 

' 2.78 

91 

92 

■ 91 

.99 

1 1 

.61 

91 

98 

2.01 

91. 9r 

2.41 

91 

.96 

2.81 

92 

93 

92 

.99 

1 

.62 

92 

.98 

2.03 

92.9 

2.43 

92 

.96 

2.84 

93 

94 

93 

.99 

1 

.64 

93 

.98 

2.05 

93.97 

2.46 

93 

.96 

2.87 

! 94 

95 

94 

.99 

1 

.66 

94 

.98 

2.07 

94.97 

2.49 

94 

.96 

1 2.90 

96 

96 

95 

99 

1 

.68 

95 

.98 

2.09 

95.97 

2.51 

95 

.96 

2.94 

90 

97 

96 

99 

1 

.69 

96 

.98 

2.12  1 

96.97 

2.54 

96 

.95 

2.96 

97 

98 

97 

.99 

1 

.71 

97 

.98 

2.  14  j 

97.97 

2.57 

97 

.95 

2.98 

! » 

99 

98 

.98 

l 

.73 

98 

.98 

2.16 

98.97 

2.59 

98 

.95 

3 07 

91 

!00 

99 

.98 

1 

.75 

99 

.98 

2.18 

99.97 

2.62 

99 

.95 

3 . 06 

100 

Distance. 

Dep. 

i ^ 

.a  t 

Dep. 

Lat. 

Dep 

Lat. 

E 

>ep. 

Ln* 

i ^ 

' 1 

| 

it 

89  Deg. 

88|  Deg. 

88 1 

Deg. 

88}  Deg 

76 


TRAVERSE  TA  RLE. 


Distance.; 

2 Deg. 

24  Deg. 

Deg. 

2$  Deg. 

Distance.! 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep 

1 

1.00 

0 

03 

J .00 

0.04 

“Too 

0, 

.04 

1 

.00 

0.05 

1 

2 

2.0C 

0 

07 

2.00 

0.08 

2.00 

0, 

.09 

2 

.00 

0.10 

2 

3 

3.0C 

0 

10 

3.00 

0.12 

3.00 

3. 

.13 

3, 

.00 

0.14 

3 

1 

4.0C 

0 

14 

4.00 

0.16 

4.00 

i 0. 

.17 

4. 

00 

0.19 

4 

5 

6.0C 

0 

17 

5.00 

0.20 

5 . 00 

0, 

,22 

4, 

.99 

0.24 

5 

8 

6.0C 

0 

21 

6.00 

0.24 

5.99 

0, 

.26 

5, 

.99 

0.29 

6 

7 

7.00 

0 

24 

6.99 

0.27 

6.99 

1 0, 

,31 

6, 

.99 

0.34 

7 

8 

7.99 

0 

28 

7.99 

0.31 

7.99 

| 0. 

.35 

7, 

.99 

0.38 

i 8 

9 

8.99 

0 

31 

8.99 

0.36 

8.99 

0. 

.39 

8, 

.99 

0.43 

9 

10 

9.99 

0 

35 

9.99 

0 39 

9.99 

0, 

.44 

9 

.99 

0.48 

1 io 

11 

10.99 

0 

38 

10.99 

0.43 

10.99 

| 0. 

.48 

10, 

.99 

0.53 

11 

12 

1 1.99 

0 

42 

1 1 . 99 

0.47| 

11.99 

0. 

.52 

11, 

.99 

0.58 

12 

13 

12.99 

0 

45 

1 2 . 99 

0.51 

12.99 

0. 

.57 

12, 

.99 

0.62 

13 

14 

13.99 

0 

49 

13.99 

0.55 

13.99 

0. 

,61 

13, 

.98 

0.67 

14 

15 

14.99 

0 

52 

14.99 

0.59 

14.99 

0. 

,65 

14, 

.98 

0.72 

15 

16 

15.99 

0 

56 

15.99 

0.63 

15.99 

0. 

,70 

15, 

.98 

0.77 

16 

17 

16.99 

0 

59 

16.99 

0.67 

16.98 

0. 

,74 

16, 

.98 

0.82 

17 

18 

17.99 

0 

63 

17.99 

0.71 

17.98 

1 0. 

,79 

17, 

.98 

0.86 

18 

19 

18.99 

0 

66 

18.99 

0 . 75 

18.98 

i °- 

.83 

18, 

.98 

0.91 

1 19 

20 

19.99 

0 

70 

19.98 

0.79 

19.98 

1 0. 

.87 

19, 

.98 

0.96 

1 20 

21 

20.99 

0 

73 

20.98 

6.82 

20.98 

0. 

92 

20, 

.98 

1.01 

f2! 

22 

21.99 

0, 

.77 

21.98 

0.86 

21.98 

i o, 

,96 

2i, 

.97 

1.06 

22 

23 

22.99 

0, 

.80 

22.98 

0.90 

22.98 

1 1, 

.00 

22, 

.97 

1.10 

23 

24 

23.99 

0. 

.84 

23.98 

0.94 

23.98 

1. 

.05 

23, 

.97 

1.15 

I 24 

25 

24.98 

0. 

.87 

24.98 

0.98 

24.98 

1, 

,09 

24, 

.97 

1.20 

25 

26 

25. 9H 

0, 

.91 

25.98 

1.02 

25.98 

1, 

.13 

25, 

.97 

1.25 

26 

27 

26.98 

0, 

.94 

26.98 

1.06 

26.97 

1, 

.18 

26, 

.97 

1.30 

27 

28 

27.98 

0, 

.98 

27.98 

1.10 

27.97 

1 , 

.22 

27, 

.97 

1.34 

28 

29 

28.98 

1. 

.01 

28.98 

1.14 

28.97 

1. 

,26 

28, 

.97 

1.39 

29 

30 

29.98 

1. 

.05 

29.98 

1.18 

29.97 

1, 

.31 

29. 

,97 

1 .44 

30 

31 

30.98 

1 , 

.08 

30.98 

1.22 

30.97 

1 , 

.35 

30, 

.96 

1.49 

31 

32 

31.98 

i! 

.12 

31.98 

1.26 

31 .97 

1. 

,40 

31, 

.96 

1.54 

32 

33 

32.98 

i , 

.15 

32.97 

1.30 

32.97 

1. 

,44 

32, 

.96 

1.58 

33 

34 

33.98 

i , 

.19 

33.97 

1.33 

33.97 

1. 

,48 

33, 

.96 

1.63 

34 

35 

34.98 

i , 

.22 

34.97 

1.37 

34.97 

1 , 

,53 

34, 

.96 

1.68 

35 

36 

35 . 98 

i, 

.26 

35.97 

1.41 

35.97 

1 , 

,57 

35, 

.96 

1.73 

36 

37 

36.98 

i , 

.29 

36.97 

1.45 

36.96 

1 , 

,61 

36, 

.96 

1.78 

37 

38 

37.98 

i 

.33 

37.97 

1.49 

37.96 

1. 

,66 

37, 

.96 

1.82 

38 

39 

38.98 

i 

.36 

38.97 

1.53 

38.96 

1 , 

,70 

38, 

.96 

1.87 

39 

40 

39.98 

i 

.40 

39.97 

167 

39.96 

1 . 

.75 

39, 

.95 

1.92 

40 

11 

40.98 

i 

.43 

40.97 

1 61 

40.96 

1 , 

,77 

40, 

.95 

1 97 

41 

42 

43  97 

i 

4~ 

41.97 

1 .65 

41.96 

1. 

.83 

41, 

,95 

2.02 

42 

43 

42.97 

t ^ ' 

.50 

42.97 

1.69 

42.96 

1. 

,88 

42, 

.95 

2.06 

43 

44 

43.97 

54 

43.97 

1.73 

43.96 

1 , 

,92 

43, 

,95 

2.11 

44 

46 

44.97 

i , 

.57 

44.97 

1.77 

44.96 

1 . 

,96 

44. 

,95 

2.16 

45 

46 

45.97 

i 

.61 

45.96 

1.81 

45.96 

2. 

,01 

45. 

,96 

2.21 

46 

47 

1 46.97 

l 

.64 

46.96 

1.85 

46.96 

2. 

,05 

46. 

,95 

2.25 

47 

48 

47.97 

i , 

.68 

47.96 

1.88 

47.95 

2. 

.09 

47, 

.95 

2.30  | 

48 

49 

! 48.97 

i. 

.71 

48.96 

1.92 

48.95 

2. 

,14 

48, 

.94 

2.35 

49 

60 

49.97 

i 

.74 

49.96 

1.96 

49.95 

2. 

18 

49. 

94 

2.40 

60 

<u 

o 

C 

as 

u 

5 

1 Dep 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lut. 

Distance. 

83  Deg 

87$  Deg. 

87^ 

Deg. 

87*  Deg. 

c 

& 

r* 

a 

L 

"61 

52 

53 

54 

65 

56 

Cnr 

6<? 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

|00 

8 

I 

at 

5 


TRAVERSE  TABLE. 


77 


eg- 

Deg. 

2* 

Deg. 

Deg. 

O 

5* 

? 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

§ 

1.78 

50.96 

2.00 

50.95 

2.22 

50.94 

2.45 

' 51 

1 .81 

51.96 

2.04 

51.95 

2.27 

51.94 

2 60 

52 

1.85 

52.96 

2.08 

52.95 

2.31 

! 52.94 

2.54 

53 

1.88 

53.96 

2.12 

53.95 

2.36 

53.94 

2.59 

i 54 

1.92 

54.96 

2.16 

54.95 

2.40 

54.94 

2 64 

! 65 

1.95 

55.9c 

, 2.20 

55.95 

2.44 

55.94 

2 69 

56 

1 99 

56 . 9# 

2.24 

56.95 

2.49 

56.93 

2.73 

5? 

2.02 

57.96 

2.28 

57.94 

2.53 

57.93 

2.78 

58 

2.06 

58.95 

2.32 

58.94 

2.57 

58.93 

2.83 

I 59 

2.09 

59.95 

2.36 

59.94 

2.62 

69.93 

2.88 

| 60 

2. 13 

60,95 

2.39 

60.94 

2.66 

60.93 

2.93 

1 B1 

2.16 

61.95 

2.43 

61.94 

2.70 

61.93 

2.97 

1 62 

2.20 

62.95 

2.47 

62.94 

2.75 

62.93 

3.02 

1 63 

2.23 

63.95 

2.51 

63.94 

2.79 

63.93 

3.07 

64 

2.27 

64.95 

2.55 

64.94 

2.84 

64.93 

3. 12 

65 

2.30 

65.95 

2.59 

65.94 

2.88 

65.92 

3.17 

66 

2.34 

66.95 

2.63 

66.94 

2.92 

66.92 

3.21 

67 

2.37 

67.95 

2.67 

67.94 

2.97 

67.92 

3.26 

68 

2.41 

68.95 

2.71 

68.93 

3.01 

68.92 

3.31 

69 

2.44 

69.95 

2.75 

69.93 

3.05 

69.92 

3.36 

70 

2.48 

70.95 

2.79 

70.93 

3.10 

70.92 

3.41 

1 71 

2.51 

71.94 

2.83 

71.93 

3.14 

71.92 

3.45 

1 72 

2.55 

72.94 

2.87 

72.93 

3.18 

72.92 

3.50  1 

1 73 

2.58 

73.94 

2.91 

73.93 

3.23 

73.91 

3.55  1 

74 

2.62 

74.94 

2.94 

74.93 

3.27 

74.91 

3.60 

75 

2.65 

75.94 

2.98 

75.93 

3.31 

75.91 

3.65 

76 

2.69 

76.94 

3.02 

76.93 

3.36 

76.91 

3.70 

77 

2.72 

77.94 

3.06  ! 

77.93 

3.40 

77. 9J 

3.74 

78 

2.76 

78.94 

3.  JO 

78.92 

3.45 

78.91 

3.79 

79 

2.79 

79.94  ! 

3.14 

79.92 

3.49 

79.91 

3.84 

80 

2.83 

80.94  ! 

3.18 

80.92 

3.53 

80.91 

3.89 

'81 

2.86 

81.94 

3.22 

81.92 

3.58 

81.91 

3.93 

82 

2.90 

82.94  ! 

3.26 

82.92 

3.62 

82.90 

3.98 

83 

2.93 

83.94 

3.30 

83.92 

3.66 

83.90 

4.03 

84 

2.97 

84.93 

3.34 

84.92 

3.71 

84.90 

4.08 

85 

3.00 

85.93 

3.38 

85.92 

3.75 

8d  9e 

4.13 

86 

3.04 

86.93 

3.42 

86.92 

3.79 

86.90 

4.17 

87 

3.07 

87.93 

3.45 

87.92 

3.84 

87.90 

4.22 

88 

3.11 

88.93 

3.49 

88.92 

3.88 

88.90 

4.27 

89 

3.14 

89.93 

3.53 

89.91 

3.93 

89.90 

4.32 

90 

3.18 

90.93 

3.57 

90.91 

3.97 

90.90 

4.37 

91 

3.21 

91.93 

3.61 

91.91 

4.01 

91.89 

4.41 

92 

3.25 

92.93 

3.65 

92.91 

4.06 

92.89 

4.46 

93 

3.28 

93.93 

3.69 

93.91 

4.10  1 

93.89 

4.51 

94 

3.32 

94.93 

3.73 

94.91 

4.14 

94.89 

4.56  1 

95 

3 35 

95.93 

3.77 

95.91 

4.19 

95.89 

4.61 

96 

3.39 

96.93 

3.81 

96.91 

4.23 

96.89 

4.65  ! 

97 

3.42 

97.92 

3.85 

97.91 

4.27 

97.89 

4.79 

98 

3.46 

98.92 

3.89 

98.91 

4.32 

98.89 

4.75 

99 

3.49 

99.92 

3.93 

99.91 

4.36 

99.88 

4.80j 

iOO 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Distonce.j 

°g- 

87|  Deg. 

87J  Deg. 

872  Deg. 

78 


TRAVERSE  TAHLE. 


c 

s* 

3 Deg 

3?  Deg 

3^  Deg. 

3?  Deg. 

! t 

p* 

3 

O 

© 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep 

i P 

1 

1.00 

0.05 

1.00 

0.06 

1.00 

0 .06 

“iVoo- 

0.06 

1 ‘ i 

2 

2.00 

0.  10 

2.00 

0.11 

2.00 

0.12 

2.00 

0 13 

I 2 

1 3 

3.00 

0. 16 

3.00 

0.17 

2.99 

0.18 

2.99 

9.20 

3 

4 

3.99 

0.21 

3.99 

0.23 

3.99 

0.24 

3.99 

0.26 

4 

fi 

4.99 

0.26 

4.99 

0.28 

4.99 

0.31 

4.99 

0.33 

5 

0 

5.99 

0.31 

5.99 

0.34 

5.99 

0.37 

5.99 

0.39 

6 

7 

6.99 

0.37 

6.99 

0.40 

6.99 

0.43 

6.99 

0.46 

7 

0 

7.99 

0.42 

7.99 

0.45 

7.99 

0.49 

7.98 

0.62 

8 

9 

8.99 

0.47 

8.99 

0.51 

8.98 

0.55 

8.98 

0.59 

9 

10 

9.99 

0.52 

9.98 

0.57 

9.98 

0.61 

9.98 

0.65 

10 

11 

10.98 

0 . 58 

10.98 

0.62 

10.98 

0.67 

10.98 

0.72 

11 

12 

11.98 

0.63 

11.98 

0.68 

11.98 

0.73 

11.97 

0.78 

12 

13 

12.98 

0.68 

12.98 

0.73 

12.98 

0.79 

12.97 

0.85 

13 

14 

13.98 

0.73 

13.98 

0.79 

13.97 

0.85 

13.97 

0.92 

14 

15 

14.98 

0.79 

14.98 

0.85 

14.97 

0.92 

14.97 

0.98 

15 

16 

15.98 

0.84 

15.97 

0.91 

15.97 

0.98 

15.97 

1.05 

16 

17 

16.98 

0.89 

16.97 

0.96 

16.97 

1.04 

16.96 

1 .11 

17 

18 

17.98 

0.94 

17.97 

1.02 

17.97 

1.10 

17.96 

1.18 

18 

19 

18.98 

0.99 

18.97 

1.08 

18.96 

1.16 

18.96 

1.24 

19 

20 

19.97 

1.05 

19.97 

1.13 

19.96 

1.22 

19.96 

1.31 

20 

21 

20.97 

l .10 

20.97 

1.19 

20.96 

1.28 

20.96 

1.37 

21 

22 

21.97 

1.15 

21 .96 

1.25 

21.96 

1.34 

21.95 

1.44 

22 

23 

22.97 

1.20 

22.96 

1.30 

22.96 

1.40 

22.95 

1.50 

23 

24 

23 . 97 

1.26 

23.96 

1.36 

23.96 

1.47 

23.95 

1.57 

24 

25 

24.97 

1.31 

24.96 

1.42 

24.95 

1.53 

24.95 

1.64 

25 

26 

25.96 

1.36 

25.96 

1.47 

25.95 

1.59 

25.94 

1.70 

26 

27 

26.96 

1.41 

26.96 

1.53 

26.95 

1.65 

26.94 

1.77 

27 

28 

27.96 

1.47 

27.95 

1.59 

27.95 

1.71  | 

27.94 

1.83 

28 

29 

28.96 

1.52 

28.95 

1.64 

28.95 

1.77 

28.94 

1.90 

29 

30 

29.96 

1.57 

29.95 

1.70 

29.94 

1.83 

29.94 

1.96 

30 

31 

30.96 

1 .62 

30.95 

1.76 

30.94 

1.89 

30.93 

2.03 

31 

32 

31.96 

1.67 

31.95 

1.81 

31.94 

1.95 

31.93 

2.09 

32 

33 

32.95 

1.73 

32.95 

1.87 

32.94 

2.01 

32.93 

2.16 

33 

34 

33 . 95 

1.78 

33.95 

1.93 

33.94 

2.08 

33.93 

2.22 

34 

35 

34.95 

1.83 

34.94 

1.98 

34.93 

2.14 

34.92 

2.29 

35 

36 

35.95 

1.88 

35.94 

2.04 

35.93 

2.20 

35.92 

2.35 

36 

37 

36.95 

1.94 

36.94 

2.10 

36.93 

2.26 

36.92 

2.42 

37 

38 

37.95 

1.99 

37.94 

2.15 

37.93 

2.32 

37.92 

2.49  1 

38 

39 

38.95 

2.04 

38.94 

2.21 

38.93 

2.38 

38.92 

2.55  ! 

39 

40 

39.95 

2.09 

39.94 

2.27 

39.93 

2.44 

39.91 

2.62  | 

40 

41  1 

40  94 

2. 15 

40.93 

2.32 

40.92 

2.50 

40.91 

2.68 

41 

42  | 

41.94 

2.20 

41.93 

2.38 

41.92 

2.56 

41.91 

2.75  1 

<2 

43  1 

42.94 

2.25 

42.93 

2.44 

42.92 

2.63 

42.91 

2.81  1 

43 

44  | 

43.94 

2.30 

43 . 93 

2.49 

43.92 

2.69 

43.91 

2.88 

44 

45  1 

44.94 

2.36 

44.93 

2.55 

44.92 

2.75 

44.90 

2.94 

45 

46 

45.94 

2.41 

45.93 

2.61 

45.91 

2.81 

45.90 

3.01 

16 

47 

46.94 

2.46 

46.92 

2.66 

46.91 

2.87 

46.90 

3.07 

47 

48  , 

47.93 

2.51 

47.92 

2.72 

47.91 

2.93 

47.90 

3.14 

48 

49  1 

48.93 

2.56 

48.92 

2.78 

48.91 

2.99 

48.90 

3.20 

19 

50 

49.93 

2.62 

49.92 

2.83 

49.91 

3.05 

49.89 

3.27 

50 

V 

o 

a 1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

® 

§ 

cd 

| | 
O j 

1 

87  Deg. 

86?  Deg. 

00£  Deg. 

86?  Deg. 

3 

Ti 

Q 

TRAVERSE  TABLE. 


79 


9 

1 

8 

3 Deg. 

34  Deg. 

3s  Deg. 

3|  Deg. 

Distance.] 

1 Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

50.93 

2.67 

50.92 

2.89 

"50.90 

Til 

‘50789 

3.34 

51 

52 

51 .93 

2.72 

51 .92 

2.96 

51.90 

3.17 

51.89 

3.40 

52 

53 

52.93 

2.77 

52.91 

3.00 

52.90 

3.24 

52.89 

3.47 

53 

54 

53.93 

2.83 

53.91 

3.06 

53.90 

3.30 

53.88 

3.53 

54 

55 

54.92 

2.88 

54.91 

3.12 

54.90 

3.36 

54.88 

3.60 

55 

5 f 

55.92 

2.93 

55  91 

3.17 

55.90 

3.42 

55.88 

3.66 

56 

67 

56.92 

2.98 

56.91 

3.23 

56.89 

3.48 

56.88 

3 73 

57 

58 

57.92 

3.04 

57.91 

3.29 

57  89 

3.54 

57.88 

3.79 

58 

59 

58.92 

3.09 

58.91 

3.34 

58.89 

3.60 

58-871  3.86 

59 

60 

59 . 92 

3.14 

59.90 

3.40 

59.89 

3.66 

59.87 

3.92 

60 

61 

60.92 

3.19 

60.90 

3.46 

60.89 

3“  72 

60.87 

3.99 

61 

62 

61.92 

3.24 

61.90 

3.51 

61.88 

3.79 

61.87 

4.05 

62 

63 

62.91 

3.30 

62.90 

3.57 

62.88 

3.85 

62.87 

4.12 

63 

64 

63.91 

3.35 

63.90 

3.63 

63.88 

1 3.91 

63.86 

4.19 

64 

65 

64.91 

3.40 

64.90 

3.69 

64.88 

3.97 

64.86 

4.25 

65 

60 

65.91 

3.45 

65.89 

3.74 

65.88 

4.03 

65.86 

4.32 

66 

67 

66.91 

3.51 

66.89 

3.80 

66.88 

4-09 

66.86 

4.38 

67 

08 

67.91 

3.56 

67.89 

3.86 

67.87 

4.15 

67.85 

4.45 

68 

69 

68.91 

3.61 

68.89 

3.91 

68.87 

4.21 

68.85 

4.51 

69 

70 

69.90 

3.66 

69.89 

3.97 

69.87 

4.27 

69.85 

4.58 

70 

71 

70.90 

3.72 

70.89 

4.03 

70.87 

I 4.33 

70.85 

4.64 

71 

72 

71.90 

3.77 

71.68 

4.08 

71.87 

4.40 

71.85 

4.71 

72 

73 

72.90 

3.82 

72.88 

4.14 

72.86 

4.46 

72.84 

4.77 

73 

74 

73.90 

3.87 

73.88 

4.20 

73.86 

4.52 

73.84 

4.84 

74 

75 

74 . 90 

3.93  1 

74.88 

4.25 

74.86 

4.58 

74.84 

4.91 

75 

76 

75.90 

3.98 

75.88 

4.31 

75  86 

4.64 

75.84 

4.97 

76 

77 

76.89  j 

4.03 

76.88 

4.37 

76.86 

4.70 

76.84 

5.04 

77 

78 

77.89 

4.08 

77 . 8»7 

4.42 

77.85 

4.76 

77.83 

5.10 

78 

79 

78.89  ! 

4.13 

78.87 

4.48 

78.85 

4.82 

78.83 

5.17 

79 

80 

79.89  i 

4.19 

79.87 

4.54 

79.85 

4.88 

79.83 

5.23 

80 

81 

80.89 

4.24 

80.87 

' 4.59 

80.85 

4.94 

80.83 

5.30 

81 

82 

81.89 

4.29 

81.87 

4.65 

81.85 

5.01 

81.82 

5.36 

82 

83 

82.89  ; 

4.34 

82.87 

4.71 

82.85 

5.07 

82.82 

5.43 

83 

84 

83.88  1 

4.40 

83.86 

476 

83.84 

5.13 

83.82 

5.49 

84 

85 

84.88  | 

4.45 

84.86 

4.82 

84.84 

5.19 

84.82 

5.56 

85 

86 

85.88  | 

4.50 

85.86  ; 

488 

85.84 

5.25 

85.82 

5.62 

86 

87 

86.88 

4.55 

86.86  | 

4.93 

86.84 

5.31 

86.81 

5.69 

87 

88  ! 

87.88 

4.61 

87.86  ! 

4.99 

87.84 

5.37 

87  .81 

5.76 

88 

89  ! 

88.88 

4.66 

88.86  1 

5.05 

88.83 

5.43 

1 88.81 

5.82 

89 

90  | 

89.88 

4.71 

89.86 

5.10 

89.83 

5.49 

89.81 

5.89 

90 

91  | 

90.88 

4.76 

90.85 

5.16 

90.83 

5.56 

90.81 

5.95 

91 

92 

91.87 

4.81 

91.85 

5.22 

91.83 

5.62 

91.80 

6.02 

92 

93 

92.87 

4.87 

92.85 

5.27 

92.83 

5.68 

92.80 

6.08 

93 

94 

93.87 

4.92  ! 

93.85 

5.33 

93.82 

5.74 

93.80 

6.15 

94 

96 

94.87 

4.97  1 

94.85 

5.39 

94.82  1 

5.80 

94.80 

6.21 

95 

9f 

95.87 

5.02 

95.85  . 

5.44 

95.82  i 

5.86 

95.79 

6.28 

96 

97 

96.87 

5.08 

96.84 

6 50 

96.82 

5.92 

96.79 

6.34 

97 

98 

97.87 

5.13 

97.84 

6.56 

97.82 

5.98 

97.79 

6.41 

98 

99 

98.86 

5.18 

98.84  ; 

5.61 

98.82 

6.04 

98.79 

6.47 

99 

IOC 

99.80  1 

5.23 

99.84  ' 

5.67 

99.81 

6.10 

99.79 

6.54 

100 

© 

o 

s 

Dop. 

Lat. 

Dep.  | 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

©‘ 

1 

ri 

"S 

5 | 

87  Deg. 

861  Deg. 

86  i ] 

Deg. 

863  Deg. 

d 

cr 

5 

80 


TRAVERSE  TABLE 


D 

5 

ST 

4 Deg. 

4*  Deg. 

n 

4-2  Dog. 

4$  Deg. 

3 

r 

c 

b 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  | 

Dep. 

b 

8 

1 

1.00 

0.07 

77oo 

0.07 

Too 

0.08 

Too 

6708 

~1 

a 

2.00 

0.14 

1.99 

0.15 

1.99 

0 16 

1.99 

0.17 

2 

3 

2.99 

0.21 

2.99 

0.22 

2.99 

0 24 

2.99 

0.25 

3 

4 

a 99 

0.28 

3.99 

0.30 

3.99 

0 31 

3.98 

0.33 

4 

6 

4.  99 

0.35 

4.99 

0.37 

4.98 

0.39 

4.98 

0.41 

5 

0 

5.99 

0.42 

5.98 

0.44 

5.98 

0.47 

5.98 

0.60 

6 

7 

6.98 

0.49 

6.98 

0.52 

6.98 

0.55 

6.97 

0.58 

7 

8 

7.98 

0.56 

7.98 

0.59 

7.98 

0.63 

7.97 

0.66 

6 

9 

8.98 

0.63 

8.98 

0.67 

8.97 

0.71 

8.97 

0.75 

0 

10 

9.98 

0.70 

9.97 

0.74 

9.97 

0.78 

9.97 

0.83 

10 

11 

10.97 

0.77 

10.97 

0.82 

10.97 

0.86 

10.96 

0.91 

11 

12 

11.97 

0.84 

11.97 

0.89 

11.96 

0.94 

11.96 

0.99 

12 

13 

12.97 

0.91 

12.96 

0.96 

12.96 

1.02 

12.96 

1.08 

13 

14 

13.97 

0.98 

13.96 

1.04 

13.96 

1.10 

13.95 

1.16 

14 

15 

14.96 

1 .05 

14.96 

1.11 

14.95 

1.18 

14.95 

1.24 

15 

16 

15.96 

1.12 

15.96 

1.19 

15.95 

1.26 

15.95 

1.32 

16 

17 

16.96 

1.19 

16.95 

1.26 

16.95 

1.33 

16.94 

1.41 

17 

18 

17.96 

1.26 

17.95 

1.33 

17.94 

1.41 

17.94 

1 .49 

>18 

19 

18.95 

1.33 

18.95 

1.40 

18.94 

1.49 

18.93 

1.57 

19 

20 

19.95 

1.40 

19.95 

1.48 

19.94 

1.57 

19.93 

1.66 

20 

21 

20.95 

1.46 

20.94 

1.56 

20.94 

1.65 

20.93 

1.74 

21 

22 

21.95 

1.53 

21.94 

1.63 

21.93 

1.73 

21.92 

1.82 

22 

23 

22.94 

1.60 

22.94 

1.70 

22.93 

1.80 

22.92 

1.90 

23 

24 

23.94 

1.67 

23.93 

1.78 

23.93 

1.88 

23.92 

1.99 

24 

25 

24.94 

1.74 

24.93 

1.85 

24.92 

1.96 

24.91 

2.07 

25 

26 

25.94 

1.81 

25.93 

1.93 

25.92 

2.04 

25.91 

2.15 

26 

27 

26.93 

1.88 

26.93 

2.00 

26.92 

2.12 

26.91 

2.24 

27 

28 

27.93 

1.95 

27.92 

2.08 

27.91 

2.20 

27.90 

2.32 

28 

29 

28.93 

2.02 

28.92 

2.15 

28.91 

2.28 

28.90 

2.40 

29 

30 

29.93 

2.09 

29.92 

2.22 

29.91 

2.35 

29.90 

2.48 

30 

31 

30.92 

i 2.16 

30.91 

2.30 

30.90 

2.43 

30.89 

2.57 

'31 

32 

31.92 

2.23 

31.91 

2.37 

31.90 

2.51 

31.89 

2.65 

82 

33 

32.92 

| 2.30 

32.91 

2.45 

32.90 

2.59 

32.89 

2.73 

33 

34 

33.92 

1 2.37 

33.91 

2.52 

33.90 

2.67 

33.88 

2.82 

34 

35 

34.91 

| 2.44 

34.90 

2.59 

34.89 

2.75 

34.88 

2.90 

35 

36 

35.91 

1 2.51 

35.90 

2.67 

35.89 

2.82 

35.88 

2.98 

36 

37 

36.91 

2.58 

36.90 

2.74 

36.89 

2.90 

36.  §7 

3.06 

37 

38 

37.91 

2.65 

37.90 

2.82 

37.88 

2.98 

37.87 

3.15 

38 

39 

38.90 

2.72 

38.89 

2.89 

38.88 

3.06 

38.87 

3.23 

39 

40 

39.90 

2.79 

39.89 

2.96 

39.88 

3.14 

39.86 

3.31 

40 

41 

40.90 

2.86 

40.89 

3.04 

40.87 

3.22 

40.86 

3.40 

41 

42 

41.90 

2.93 

41.88 

3.11 

41.87 

3.30 

• 41.86 

| 3.48 

12 

43 

42.90 

3.00 

42.88 

3.19 

42.87 

3.37 

42.85 

1 3.56 

13 

44 

43.89 

3.07 

43.88 

3.26 

43.86 

3.45 

43.85 

3.64 

44 

45 

44.89 

3.14 

44.88 

3.33 

44.86 

3.53 

44.85 

3.73 

4f 

46 

45.89 

3.21 

45.87 

3.41 

45.86 

3.61 

45.84 

3.81 

16 

47 

1 46.89 

3.28 

46.87 

3.48 

46.86 

3.69 

46.84 

3.89 

47 

48 

47.88 

3.35 

47.87 

3.56 

47.85 

3.77 

47.84 

3.97 

48 

49 

1 48.88 

3.42 

48.87 

3.63 

48.85 

3.84 

48.83 

4.06 

49 

50 

| 49.88 

3 49 

49.86 

3.71 

49.85 

3.92 

49 .83 

4.14 

50 

' 8 
q 

Dep. 

1 Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

c 

1 

Q 

86  Deg. 

85$  Deg. 

85^  Dog. 

85$  Deg. 

2 

CD 

5 

TRAVERSE  TABLE. 


81 


Distance*! 

4 Deg. 

4i  Deg. 

4*1 

)eg. 

4|  Deg. 

Distance.  | 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dcp. 

51 

50.88 

3.66 

50.86 

3.78 

50.84 

4.00 

50.82 

4.22 

51 

52 

51.87 

3.63 

51.86 

3.85 

51.84 

4.08 

51.82 

4.31 

52 

53 

52.87 

3.70 

52.85 

3.93 

52.84 

4.16 

52.82 

4.39 

53 

54 

53.87 

3.77 

53.85 

4.00 

53.83 

4.24 

53.81 

4.47 

54 

55 

54.87 

3.84 

54.85 

4.08 

54.83 

4.32 

54.81 

4 55 

55 

56 

55.86 

3.91 

55.85 

4.15 

55.83 

4.39 

55.81 

4.64 

56 

57 

56.86 

3.98 

56.84 

4.22 

56.82 

4.47 

56.80 

4.72 

57 

58 

57.86 

4.05 

57.84 

4.30 

57.82 

4.55 

57  80 

4.80 

58 

69 

58.86 

4.12 

58.84 

4.37 

58.82 

4.63 

58.80 

4.89 

59 

60 

59.85 

4.19 

59.84 

4.45 

59.82 

4.71 

59.79 

4.97 

60 

61 

60.85 

4.26 

60.83 

4.52 

60.81 

4.79 

60.79 

5.05 

61 

62 

61.85 

4.32 

61.83 

4.59 

61.81 

4.86 

61.79 

5.13 

62 

63 

62.85 

4.39 

62.83 

4.67 

62.81 

4.94 

62.78 

5.22 

63 

64 

63.84 

4.46 

63.82 

4.74 

63.80 

5.02 

63.78 

5.30 

64 

65 

64.84 

4.53 

64.82 

4.82 

64.80 

5.10 

64.78 

5.38 

65 

66 

65.84 

4.60 

65.82 

4.89 

65.80 

5.18 

65.77 

5.47 

66 

67 

66.84 

4.67 

66.82 

4.97 

66.79 

5.26 

66.77 

5.55 

67 

68 

67.83 

4.74 

67.81 

5.04 

67.79 

5.34 

67.77 

5.63 

68 

69 

68.83 

4.81 

68.81 

5.11 

68.79 

5.41 

68.76 

5.71 

69 

70 

69.83 

4.88 

69.81 

5.19 

69.78 

5.49 

69.76 

5.80 

70 

71 

70.83 

4.95 

70.80 

5.26 

70.78 

5.57 

70.76 

5.88 

71 

72 

71.82 

5.02 

71.80 

5.34 

71.78 

5.65 

71.75 

5.96 

72 

73 

72.82 

5.09 

72.80 

5.41 

72.77 

5.73 

72.75 

6.04 

73 

74 

73.82 

5.16 

73.80 

5.48 

73.77 

5.81 

73.75 

6.13 

74 

75 

74.82 

5.23 

74.79 

5.56 

74.77 

5.88 

74.74 

6.21 

75 

76 

75.81 

5.30 

75.79 

5.63 

75.77 

5.96 

75.74 

6.29 

76 

77 

76.81 

5.37 

76.79 

5.71 

76.76 

6.04 

76.74 

6.38 

77 

78 

77.81 

5.44 

77.79 

5.78 

77.76 

6.12 

77.73 

6.46 

78 

79 

78.81 

5.51 

78.78 

5.85 

78.76 

6.20 

78.73 

6.54 

79 

80 

79.81 

5.58 

79.78 

5.93 

79.75 

6.28 

79.73 

6.62 

80 

81 

80.80 

5.65 

80.78 

6.00 

80.75 

6.36 

80.72 

6.71 

81 

82 

81.80 

5.72 

81.78 

6.08 

81.75 

6.43 

81.72 

6.79 

82 

83 

82.80 

5.79 

82.77 

6.15 

82.74 

6.51 

82.71 

6.87 

83 

84 

83.80 

5.86 

83.77 

6.23 

83.74 

6.59 

83.71 

6.96 

84 

85 

84.79 

5.93 

84.77 

6.30 

84.74 

6.67 

84.71 

7.04 

85 

86 

85.79 

6.00 

85.76 

6.37 

85.73 

6.75 

85.70 

7.12 

86 

87 

86 . 79 

6.07 

86.76 

6.45 

86.73 

6.83 

86.70 

7.20 

87 

88 

87. 7S 

6.14 

87.76 

6.52 

87.73 

6.90 

87.70 

7.29 

88 

89 

88.78 

6.21 

88.76 

6.60 

88.73 

6.98 

88.70 

7.37 

89 

90 

89 . 78 

6.28 

89.75 

6.67 

89 . 72 

7.06 

89.69 

7.45 

90 

91 

90.78 

6.35 

90.75 

6.74 

90.72 

7.14 

90.69 

7.54 

91 

92 

91.78 

6.42 

91.75 

6.82 

91.72 

7.22 

91.68 

7.62 

92 

93 

92.77 

6.49 

92.74 

6.89 

92.71 

7.30 

92.68 

7.70 

9? 

34 

93.77 

6.66 

93.74 

6.97 

93.71 

7.38 

93.68 

7.78 

9<* 

95 

94.77 

6.63 

I 94.74 

7.04 

94.71 

7.45 

94.67 

7.87 

95 

96 

95.77 

6.70 

95.74 

7.11 

95.70 

7.53 

95.67 

7.95 

96 

!)7  : 96.76 

6.77 

96.73 

7.19 

96.70 

7.61 

96.67 

8.03 

97 

98 

97.76 

6.84 

97.73 

7.26 

97.70 

7.69 

97.66 

8.12 

98 

99 

98.76 

6.91 

98.73 

7.34 

98.69 

7.77 

98.66 

8.20 

99 

100 

99.76 

6.98 

99.73 

7AL 

99.69 

7.85 

99.66 

8.28 

100 

© 

e 

9 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

2 

3 

2 

°l 

| 86  Deg. 

85^  Deg. 

85|  Deg. 

85^  Deg. 

3 

GO 

6 

TRAVERSE  TABLE 


82 


Distance.! 

5 Deg. 

5?  Deg. 

Deg. 

5|  Deg.  | 

j Distance. 1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  | 

1 

1 .00 

0. 

09 

1.00 

0.09 

1.00 

~o7io" 

0.99 

oTTo ' 

1 

2 

1.99 

0. 

17 

1.99 

0.18 

1.99 

0.19 

1.99 

0.20 

2 

3 

2.99 

0. 

26 

2.99 

0.27 

2.99 

0.29 

2.98 

0.30 

3 

4 

3.98 

0. 

35 

3.98 

0.37 

3.98 

0.38 

3.98 

0.40 

4 

5 

4.98 

0. 

44 

4.98 

0.46 

4.98 

0.48 

4.97 

0.50 

6 

6 

5.98 

0. 

52 

5.97 

0.55 

5.97 

0.58 

5.97 

0.60 

6 

7 

6.97 

0. 

61 

6.97 

0.64 

6.97 

0.67 

6.96 

0.70 

7 

« 

7.97 

0. 

70 

7.97 

0.73 

7.96 

0.76 

7.96 

0.80 

8 

9 

8.97 

0. 

78 

8.96 

0.82 

8 96 

0.86 

8.95 

0.90 

9 

10 

9.96 

0. 

87 

9.96 

0.92 

9.95 

0.96 

9.95 

1.00 

10 

11 

10.96 

0 

96 

10.95 

1.01 

10.95 

1 . 05 

10.94 

1.10 

11 

12 

1 1 .95 

l 

05 

11.95 

1.10 

11.94 

1.15 

11.94 

1.20 

12 

13 

12.95 

1. 

13 

12.95 

1.19 

12.94 

1.25 

12.93 

1.30 

13  ! 

14 

13.95 

1 

22 

13.94 

1.28 

13.94 

1.34 

13.93 

1.40 

14  i 

15 

14.94 

1 

31 

14.94 

1.37 

14.93 

1.44 

14.92 

1.50 

15 

16 

15.94 

1 

39 

15.93 

1.46 

15.93 

1.53 

15.92 

1.60 

16 

17 

16.94 

1 

48 

16.93 

1.56 

16.92 

1.63 

16.91 

1.70 

17 

18 

17.93 

1 

57 

17.92 

1.65 

17.92 

1.73 

17.91 

1.80 

18 

19 

18.93 

1 

66 

18.92 

1.74 

18.91 

1.82 

18.90 

1.90 

19 

20 

19.92 

1 

74 

19.92 

1.83 

19.91 

1.92 

19.90 

2.00 

20 

21 

20.92 

1 

83 

20.91 

1.92 

20 '90 

2.01 

20.89 

2.10 

21 

22 

21.92 

1. 

,92 

21.91 

2.01 

21.90 

2.11 

21.89 

2.20 

22 

23 

22.91 

2. 

.00 

22.90 

2.10 

22.89 

2.20 

22.88 

2.30 

23 

24 

23.91 

2. 

,09 

23.90 

2.20 

23.89 

2.30 

23.88 

2.40 

24 

25 

24.90 

2. 

,18 

24.90 

2.29 

24.88 

2.40 

24.87 

2.50 

25 

26 

25.90 

2. 

,27 

25.89 

2.38 

25.88 

2.49 

25.87 

2.60 

26 

27 

26.90 

2. 

,35 

26.89 

2.47 

26.88 

2.59 

26.86 

2.71 

27 

28 

27.89 

2, 

,44 

27.88 

2.56 

27.87 

2.88 

27.86 

2.81 

28 

29 

28.89 

2. 

,53 

28.88 

2.65 

28.87 

2.78 

28.85 

2.91 

29 

30 

29.89 

2. 

,61 

29.87 

2.75 

29.86 

2.88 

29.85 

3.01 

30 

31 

30.88 

2. 

.70 

30.87 

2.84 

30.86 

2.97 

30.84 

3.11 

31 

32 

31.88 

2. 

,79 

31.87 

2.93 

31.85 

3.07 

31.84 

3.21 

32 

33 

32.87 

2. 

.88 

32.86 

3.02 

32.85 

3.16 

32.83 

3.31 

33 

34 

33.87 

2, 

.96 

33.86 

3.11 

33.84 

3.26 

33.83 

3.41 

34 

35 

34.87 

3 

.05 

34.85 

3.20 

34.84 

3.35 

34.82 

3.51 

35 

36 

35.86 

3 

. 14 

35.85 

3.29 

35.83 

3.45 

35.82 

3.61 

36 

37 

36.86 

3 

.22 

36.84 

3.39 

36.83 

3.55 

36.81 

3.71 

37 

38 

37.86 

3 

.31 

37.84 

3.48 

37.83 

3.64 

37.81 

3.81 

38 

39 

38.85 

3 

.40 

38.84 

3.57 

38.82 

3.74 

38.80 

3.91 

39 

40 

39.85 

3 

.49 

39.83 

3.66 

39.82 

3.83 

39.80 

4.01 

40 

'41 

40.84 

3 

.57 

40.83 

3.75 

40.81 

3.93 

40.79 

4.11 

41 

42 

41.84 

3 

.66 

41.82 

3.84 

41.81 

4.03 

41.79 

4.21 

42 

43 

42  84 

3 

.75 

42.82 

3.93 

42.80 

4.12 

42.78 

4.31 

43 

44 

43.83 

3 

.83 

43.82 

4.03 

43.80 

4.22 

43.78 

4 41 

44 

45 

44.83 

3 

.92 

44.81 

4.12 

44.79 

4.31 

44.77 

4 51 

45 

10 

45.82 

4 

.01 

45.81 

4.21 

45.79 

4.41 

45.77 

4 61 

46 

17 

46.82 

4 

.10 

46.80 

4.30 

46.78 

4.50 

46.76 

4.71 

47 

48 

47.82 

4 

.18 

47.80 

4.39 

47.78 

4.60 

47.76 

4.81 

48 

49 

48.81 

4 

.27 

48,79 

4.48 

48.77 

4.70 

48.75 

4.91 

49 

50  1 49.81 

4 

.36 

49.79 

4.58 

49.77 

4.79 

49.75 

5.01 

50 

8 

| 

9 

A 

Dop. 

1 Lat. 

Dep. 

| Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

1 

2 

5 

85  Deg. 

84?  Dog. 

84^ 

Deg. 

84?  Deg. 

<oo«o«o«c«e«o*|OOoaoooao'Xaoaoocooao-^^-^^i^3<5^3-<!<}i<fOsoja»a>a»e»o>C5C»ocno*o>oi©icnoicn  ex|.Lnnnn,m 
-vjotn^-WK/^-i  o<cooviojoi^toto*— 1 ocoax!a5Ui^oo^>— I ooaiv'oiO'^MM^Io<cai^a«i^UMl-|  ouuujbiq 


TRAVERSE  TAELli. 


83 


5 Deg. 

54  Deg. 

Deg. 

5?  Deg 

I c 

I I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 

1? 

50.81 

4.44 

50.79 

4.67 

50.77 

4.89 

50  74 

5.H 

5 

51 .80 

4.53 

51.78 

4.76 

51.76 

4.98 

51  74 

5.21 

52 

52.80 

4.62 

52.78 

4.85 

52.76 

5.08 

52  73 

5.31 

53 

53.79 

4.71 

53.77 

4.94 

53.75 

5.18 

53  73 

5.41 

54 

51.79 

4.79 

54.77 

6.03 

54.75 

5.27 

54.72 

5.51 

65 

55.79 

4 88 

55.77 

5.12 

55.74 

5.37 

55.72 

5.61 

56 

56.78 

4.97 

56.76 

5.22 

56.74 

5.46 

56.71 

5.71 

57 

57.78 

5.06 

57.76 

5.31 

57.73 

5.56 

57.71 

5.81 

58 

58.78 

5.14 

58.75 

5.40 

58.73 

5.65 

58.70 

5.91 

59 

59.77 

5.23 

59.75 

5.49 

59.72 

5.75 

59.70 

6.01 

| 60 

60.77 

5.32 

60.74 

5.58 

60.72 

5.85 

60.69 

6.11 

61 

61.76 

5.40 

61.74 

5.67 

61.71 

5.94 

61.69 

6.21 

62 

62.76 

5.49 

62.74 

5.76 

62.71 

6.04 

62.68 

6.31 

63 

63.76 

5.68 

63.73 

5.86 

63.71 

6.13 

63.68 

6.41 

64 

64.75 

5.67 

64.73 

5.95 

64.70 

6.23 

64.67 

6.51 

65 

65.75 

5.75 

65.72 

6.04 

65.70 

6.33 

65.67 

6.61 

66 

66.75 

5.84 

66.72 

6.13 

66.69 

6.42 

66.66 

6.71 

6? 

67.74 

5.93 

67.71 

6.22 

67.69 

6.52 

67.66 

6.81 

68 

68.74 

6.01 

68.71 

6.31 

68.68 

6.61 

68.65 

6.91 

69 

69.73 

6.10 

69.71 

6.41 

69.68 

6.71 

69.65 

7.01 

70 

70.73 

6.19 

70.70 

6.50 

70.67 

6.81 

70.64 

7.11 

71 

71.73 

6.28 

71.70 

6.59 

71.67 

6.90 

71.64 

7.21 

72 

72.72 

6.36 

72.69 

6.68 

72.66 

7.00 

72.63 

7.31 

73 

73.72 

6.45 

73.69 

6.77 

73.66 

7.09 

73.63 

7.41 

74 

74.71 

6.54 

74.69 

6.86 

74.65 

7.19 

74.62 

7.51 

75 

75.71 

6.62 

75.68 

6.95 

75.65 

7.28 

75.62 

7.61 

76 

76.71 

6.71 

76.68 

7.05 

76.65 

7.38 

76.61 

7.71 

77 

77.70 

6.80 

77.67 

7.14 

77.64 

7.48 

77.61 

7.81 

78 

78.70 

6.89 

78.67 

7.23 

78.64 

7.57 

78.60 

7.91 

79 

79.70 

6.97 

79.66 

7.32 

79.6a 

7.67 

79.60 

8.02 

80 

80.69 

7.06 

80.66 

7.41 

80.63 

7.76 

80.59 

8.12 

81 

81.69 

7.15 

81.66 

7.50 

81.62 

7.86 

81.59 

8.22 

82 

82.68 

7.23 

82.65 

7.59 

82.62 

7.96 

82.58 

8.32 

83 

83.68 

7.32 

83.65 

7.69 

83.61 

8.05 

83.58 

8.42 

84 

84.68 

7.41 

84.64 

7.78 

84.61 

8.15 

84.57 

8.52 

85 

85.67 

7.50 

85.64 

7.87 

85.60 

8.24 

85.57 

8.62 

86 

86.67 

7.68 

86.64 

7.96 

86.60 

8.34 

86.56 

8.72 

87 

87.67 

7.67 

87.63 

8.05 

87.59 

8.43 

87.56 

8.82 

88 

88.66 

7.76 

88.63 

8.14 

88.59 

8.53 

88.55 

8.92 

89 

89.66 

7.84 

89.62 

8.24 

89.59 

8.63 

89.55 

9.02 

90 

90  65 

7.93 

90.62 

8.33 

90.58 

8.72 

90.54 

9.12 

91 

91  .65 

8.02 

91.61 

8.42 

91  58 

8.82 

91.54 

9.22 

92 

92.65 

8.11 

92.61 

8.51 

92.57 

8.91 

92.53 

9.32 

93 

93.64 

8.19 

93.61 

8.60 

93.57 

9.01 

93.53 

9.42 

94 

91.64 

8.28 

94.60 

8.69 

94.56 

9.11 

94.52 

9.52 

95 

95.63 

8.37 

95.60 

8.78 

95.56 

9.20 

95.52 

9.62  j 

96 

96.63 

8,45 

96.59 

8.88 

96.55 

9.30 

96.51 

9.72  | 

97 

97  63 

8.64 

97.59 

8.97 

97.55 

9.39 

97.51 

9.82  | 

98 

98  52 

8 63 

98.59 

9.06 

98.54 

9.49 

98.50 

9.92 

99 

1 99.62 

8.72 

99.58 

9.15 

99.54 

9.58 

99.50 

10.02 

100 

! Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

§ 

85  Dog. 

84.?  Deg. 

84£  Deg. 

84?  Deg 

J 

|o 

84 


TRAVERSE  TABLE 


Distance., 

6 Deg. 

6*  Deg. 

1 

6i  Deg. 

64  Deg. 

Distance  1 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dop. 

Lat. 

1 n 1 

Dop.  j 

1 

0.99 

0.10 

9.99 

0.11 

“0799- 

0.11 

0.99 

0712  | 

1 

8 

1 99 

0.21 

1.99 

0.22 

1.99 

0.23 

1.99 

0 24  | 

2 

3 

2 98 

0.31 

2.98 

0.33 

2.98 

0.34 

2.98 

0 35' 

3 

4 

3 98 

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52 

53  152.60 

6.46 

52.58 

6.69 

52.55 

6.92 

52.52 

7-15 

53 

54 

53.60 

6.58 

53.57 

6.81 

53.54 

7.05 

53.51 

7.28  i 

54 

55 

54.59 

6 70 

54. 50 

6.94 

54.53 

7.18 

54.50 

r.42 

55 

54i 

55.58 

6.82 

05.55 

7.07 

55.52 

7 31 

55.49 

7.55 

56 

57  ! 56.58 

6.95 

56.54 

7.19 

56.51 

7 44 

56.48 

7.69  1 

57 

58 

57.57 

7.07 

57.54 

7.32 

57.50 

7.57 

57.47 

7.82 

58 

59 

58.56 

7.19 

58.53 

7.45 

58.50 

7.70 

58.46 

7.96 

59 

60 

59.55 

7.31 

59.52 

7.57 

59.49 

7.83 

59.45 

8.09 

60 

61 

60.55 

7 43 

60.51 

7.70 

60.48 

7.96 

60.44 

8.23 

* 61 

62 

61.54 

7.56 

61.50 

7.82 

61.47 

8.09 

61.43 

8.36 

62 

63 

62.53 

7.68 

62.50 

7.95 

62.46 

8.22 

62.42 

8.50 

63 

64 

63.52 

7.80 

63.49 

8.08 

63.45 

8.35 

63.42 

8.63 

64 

65 

64.52 

7.92 

64.48 

8.20 

64.44 

8.48 

64.41 

8.77 

65 

66 

65.51 

8.04 

65.47 

8.33 

65.44 

8.61 

65.40 

8.90 

66 

67 

66  50 

8.17 

66.46 

8.46 

66.43 

8.75 

66.39 

9.04 

67 

68 

67.49 

8.29 

67.46 

8.58 

67.42 

8.88 

67.38 

9. 17 

68 

69 

68.49 

8.41 

68.45 

8.71 

68.41 

9.01 

68.37 

9.30 

69 

70 

69.48 

8.53 

69.44 

8.83 

69.40 

9.14 

69.36 

9.44 

70 

71 

70.47 

8.65 

70.43 

8.96  ' 

70.39' 

9.27 

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9.57 

71 

72 

71.46 

8.77 

: 71.42 

9.09 

71.38 

9.40 

71.34 

9.71 

72 

73 

72.46 

8.90 

72.42 

9.21 

72.38 

9.53 

72.33 

9.84 

73 

74 

73.45 

9.82 

73.41 

9.34 

73.37 

9.66 

73.32 

9.98 

74 

75 

74.44 

9.14 

74.40 

9.46 

74.36 

9.79 

74.31 

10.11 

75 

76 

75.43 

9.26 

75.39 

9.59 

75.35 

9.92 

75.31 

10.25 

76 

77 

76.43 

9.38 

76.38 

9.72 

76.34 

10.05 

76.30 

10.38 

77 

78 

77.42 

9.51 

77.38 

9.84 

77.33 

10.18 

77.29 

10.52 

78 

79 

78.41 

9.63 

78.37 

9.97 

78.32 

10.31 

78.28 

10.65 

79 

80 

79.40 

9.75 

79.36 

10.10 

79.32 

10.44 

79.27 

10.79 

80 

81 

80.40 

9.87 

80.35 

10.22 

80.31 

10.57 

80.26 

10.92 

81 

82 

81.39 

9.99 

81.34 

10.35 

81.30 

10.70 

81.25 

11.06 

82 

83 

82.38 

10.12 

82.34 

10.47 

82.29 

10.83 

82.24 

11.19 

83 

84 

83.37 

10.24 

83.33 

10.60 

83.28 

10.96 

83.23 

11.33 

84 

85 

84.37 

10.36 

84.32 

10.73 

84.27 

11.09 

84.22 

11.46 

I 85 

86 

85.36 

10.48 

85.31 

10.85 

85.26 

11.23 

85.21 

11.60 

j 86 

87 

86.35 

10.60 

86.30 

10.98 

86.26 

11.36 

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11.73 

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11.11 

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11.49 

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89 

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11.23 

88.24 

11.62 

88.19 

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! 89 

90 

89.33 

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89.28 

11.36 

89.23 

11.75 

89.18 

12.14 

| 90 

91 

90.32 

11.09 

90.27 

11.48 

90 .22 

11.88 

90.17 

12.27 

91 

92 

91.31 

11.21 

91.26 

11.61 

91.21 

12.01 

91.16 

12.41 

1 92 

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92.31 

11.33 

92.26 

11.74 

92.20 

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12.51 

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93.25 

11.86 

93.20 

12.27 

93.14 

12  68 

94 

95 

94.29 

11.58 

94.24 

11.99 

94. 19 

12.40 

94.13 

12.81 

95 

96  95.28  , 

1 1 . 70 

95.23 

12.12 

95.  18 

12.53 

95. 12 

12.95 

96 

67 

96.28 

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96.22 

12.24 

96.17 

12.66 

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13.08 

97 

98 

97.27 

11.94 

97.22 

12.37 

97.16 

12.79 

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| 13.22 

i 98 

99 

98.26 

12.07 

98.21 

12.49 

98.15 

12.92 

98. 10 

I 13.35 

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12.19 

99.20 

12.62 

99.14 

13.05 

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82£  Deg. 

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2 

1.98 

0.28 

1.98 

0.29 

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0.30 

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2.97 

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2.97 

0.43 

2.97 

0.44 

2.97 

0.46 

1 3 

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3. 96 

0.56 

3.96 

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4.95 

0.70 

4.95 

0.72 

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6 

5.94 

0.84 

5.94 

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6.93 

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6.92 

1.03 

6.92 

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7 

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7.92 

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7.91 

1.18 

7.91 

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8 

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8.91 

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8.91 

1.29 

8.90 

1.33 

8.90 

1.37 

1 9 

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9.90 

1.39 

9.90 

1.43 

9.89 

1.48 

9.88 

1.52 

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10.89 

1.53 

10.89 

1.58 

10.88 

1.63 

10.87 

1.67 

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12 

11.88 

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11.88 

1.72 

11.87 

1.77 

11.86 

1.83 

12 

13 

12.87 

1 .81 

12.87 

1.87 

12.86 

1.92 

12.85 

1.98 

13 

14 

13.86 

1.95 

13.86 

2.01 

13.85 

2.07 

13.84 

2.13 

14 

15 

14.85 

2.09 

14.85 

2.15 

14.84 

2.22 

14.83 

2.28 

15 

16 

15.84 

2.23 

15.84 

2.30 

15.82 

2.36 

15.81 

2.43 

16 

17 

16.83 

2.37 

16.83 

2.44 

16.81 

2.51 

16.80 

2.59 

17 

18 

17.82 

2.51 

17.81 

2.58 

17.80 

2.66 

17.79 

2.74 

18 

19 

18.82 

2.64 

18.80 

2.73 

18.79 

2.81 

18.78 

2.89 

19 

20 

19.81 

2.78 

19.79 

2.87 

19.78 

2.96 

19.77 

3.04 

20 

21 

20.80 

2.92 

20.78 

3.01 

20.77 

3.10 

20.76 

3. 19 

21 

22 

21.79 

3.06 

21.77 

3.16 

21.76 

3.25 

21.74 

3.35 

22 

23 

22.78 

3.20 

22.76 

3.30 

22.75 

3.40 

22.73 

3.50 

23 

24 

23.77 

3.34 

23.75 

3.44 

23.74 

3.55 

23.72 

3.65 

24 

25 

24.76 

3.48 

24.74 

3.59 

24.73 

3.70 

24.71 

3.80 

25 

26 

25.75 

3.62 

25.73 

3.73 

25.71 

3.84 

25.70 

3.96 

26 

27 

26.74 

3.76 

26.72 

3.87 

26.70 

3.99 

26.69 

4.11 

27 

28 

27.73 

3.90 

27.71 

4.02 

27.69 

4.14 

27.67 

4.26 

28 

29 

28.72 

4.04 

28.70 

4.16 

28.68 

4.29 

28.66 

4.41 

29 

30 

29.71 

4.18 

29.69 

4.30 

29.67 

4.43 

29.65 

4.56 

30 

31 

30.70 

4.31 

30.68 

4.45 

30.66 

4.58 

30.64 

4.72 

31 

32 

31.69 

4.45 

31.67 

4.59 

31.65 

4.73 

31.63 

4.87 

32 

33 

32.68 

4.59 

32.66 

4.74 

32.64 

4.88 

32.62 

5.02 

33 

34 

33.67 

4.73 

33.65 

4.88 

33.63 

5.03 

33.60 

5.17 

34 

35 

1 34.66 

4.87 

34.64 

5.02 

34.62 

5.17 

34.59 

5.32 

35 

36 

35.65 

5.01 

35.63 

5.17 

35.60 

5.32 

35.58 

5.  <8 

36 

37 

36.64 

5.15 

36.62 

5.31 

36.59 

5.47 

36.57 

5.63 

37 

38 

37.63 

5.29 

37.61 

5.45 

37.58 

5.62 

37.56 

5.78 

38 

39 

38.62 

5.43 

38.60 

5.60 

38.57 

5.76 

38.55 

5.93 

39 

40 

39.61 

5.57 

39-59 

5.74 

39.56 

5.91 

39.53 

6.08 

40 

41 

40.60 

5.71 

40.58 

5.88 

40.55 

6.06 

40.52 

6.24 

1 41 

42 

41.59 

5.85 

41.57 

6.03 

41.54 

6.21 

41.51 

6.39 

42 

43 

42.58 

6.98 

42.56 

6.17 

42.53 

6.36 

42.50 

6.54 

43 

44 

43.57 

6.12 

43.54 

6.31 

43.52 

6.50 

43.49 

6.69 

44 

45 

44  56 

6.26 

44.53 

6.46 

44.51 

6.65 

44.48 

6.85 

45 

46 

45.55 

6.40 

45.52 

6.60 

45.49 

6.80 

45.46 

7.00 

46 

47 

,46.54 

6.54 

46.5 

6.74 

46.48 

0.95 

46.45 

7.15 

47 

48 

47.53 

6.68 

47.50 

6.89 

47.47 

7.09 

47.44 

7.30 

48 

49 

48.52 

6.82 

48.49 

7.03 

48.46 

7.24 

48.43 

7.45 

49 

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6.96 

49.48 

7.17 

49.45 

7.39 

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81 J Dog. 

81$  Deg. 

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8*  Deg. 

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3 

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51 

50.50 

7.10 

50.47 

7.32 

50.44 

7.54 

50.41 

7.76 

51 

52 

51.49 

7.24 

51.46 

7.46 

51.43 

7.69 

51.39 

7.91 

52 

53 

52.48 

7.38 

52.45 

7.61 

52.42 

7.83 

52.38 

8.06 

53 

54 

53.47 

7.52 

53.44 

7.75 

53  41 

7.98 

53.37 

8.21 

54 

55 

54.46 

7.65 

54.43 

7.89 

54.40 

8.13 

54.36 

8.37 

55 

56 

55.46 

7.79 

55.42 

8.04 

55.38 

8.28 

55.35 

8.52 

56 

57 

56.45 

7.93 

56.41 

8.18 

56.37 

8.43 

56.34 

8.67 

57 

68 

57.44 

8.07 

57.40 

8.32 

57.36 

8.57 

57.32 

8.82 

58 

59 

58.43 

8.21 

58.39 

8.47 

58.35 

8.72 

58.31 

8.98 

59 

60 

59.42 

8.35 

59-38 

8.61 

59.34 

8.87 

59.30 

9.13 

60 

61 

60.41 

8.49 

60.37 

8.75 

60.33 

9.02 

60.29 

9.28 

61 

62 

61.40 

8.63 

61.36 

8.90 

61.32 

9.16 

61.28 

9.43 

62 

63 

62.39 

8.77 

62.35 

9.04 

62.31 

9.31 

62.27 

9.58 

63 

64 

63.38 

8.91 

63.34 

9.18 

63.30 

9.46 

63.26 

9.74 

64 

65 

64.37 

9.05 

64.33 

9.33 

64.29 

9.61 

64.24 

9.89 

65 

66 

65.36 

9.19 

65.32 

9.47 

65.28 

9.76 

65.23 

10.04 

66, 

67 

66.35 

9.32 

66.31 

9.61 

66.26 

9.90 

66.22 

10.19 

67 

68 

67.34 

9.46 

67.30 

9.76 

67.25 

10.05 

67.21 

10.34 

68 

69 

68.33 

9.60 

68.29 

9.90 

68.24 

10.20 

68.20 

10.50 

69 

70 

69.32 

9.74 

69.28 

10.04 

69.23 

10.35 

69.19 

10.65 

70 

71 

70.31 

9.88 

70.27 

10.19 

70.22 

10.49 

70.17 

10.80 

71 

72 

71.30 

10.02 

71.25 

10.33 

71.21 

10.64 

71.16 

10.95 

72 

73 

72.29 

10.16 

72.24 

10.47 

72.20 

10.79 

72.15 

11.10 

73 

74 

73.28 

10.30 

73.23 

10.62 

73.19 

10.94 

73.14 

11.26 

74 

75 

74.27 

10.44 

74.22 

10.76 

74.18 

11.09 

74. 13 

11.41 

75 

76 

75.26 

10.58 

75.21 

10.91 

75.17 

11.23 

75.12 

11.56 

76 

77 

76.25 

10.72 

76.20 

11.05 

76.15 

11.38 

76.10 

11.71 

77 

78 

77.24 

10.86 

77.19 

11.19 

77.14 

11.53 

77.09 

11.87 

78 

79 

78.23 

10.99 

78.18 

11.34 

78.13 

11.68 

78.08 

12.02 

79 

80 

79.22 

11.13 

79.17 

11.48 

79.12 

11.82 

79.07 

12.17 

80 

81 

80.21 

11.27 

80.16 

11.62 

80.11 

11.97 

80.06 

12.32 

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82 

81.20 

11.41 

81.15 

11.77 

81.10 

12.12 

81.05 

12.47 

82 

83 

82.19 

11.55 

82.14 

11.91 

82.09 

12.27 

82.03 

12.63 

83 

84 

83.18 

11.69 

83.13 

12.05 

83.08 

12.42 

83.02 

12.78 

84 

85 

84. 17 

11.83 

84.12 

12.20 

84.  O’7 

12.56 

84.01 

12.93 

85 

86 

85.16 

11.97 

85.11 

12.34 

85.00 

12.71 

85.00 

13.08 

86 

87 

86.15 

12.11 

86.10 

12.48 

86.04 

12.86 

85.99 

13.23 

87 

88 

87.14 

12.25 

87.09 

12.63 

87.03 

13.01 

86.98 

13.39 

88 

89 

88.13 

12.39 

88.08 

12.77 

88.02 

13.16 

87.96 

13.54 

89 

90 

89.12 

12.53 

89.07 

12.91 

89.01 

13.30 

88.95 

13.69 

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31 

90.11 

12.66 

90.06 

13.06 

90.00 

13.45 

89.94 

13.84 

91 

92 

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91.05 

13.20 

90.99 

13.60 

90.93 

14.00 

92 

93 

92.09 

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13.34 

91.98 

13.75 

91.92 

14.15 

93 

94 

93.09 

13.08 

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92.97 

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96.06 

13.50 

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14.34 

95.87 

14.76 

97 

98 

97.05 

13.64 

96.99 

14.06 

96.92 

14.49 

96.86 

14.91 

98 

99 

98.04 

13.78 

97.98 

14.21 

97.91 

14.63 

97.85 

15.06 

99 

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99.03 

13.92 

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98.90 

14.78 

98.84 

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9|  Deg 

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1.13 

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7 

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7.90 

1.25 

7.90 

1.29 

7.89 

1.32 

7.88 

, 1.35 

8 

9 

8.89 

1 .41 

8.88 

1.45 

8.88 

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8.87 

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9 

10 

9.88 

1.56 

9.87 

1.61 

9.86 

i.65 

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I 1.69 

10 

11 

10.86 

1.72 

10.86 

1.77 

10.85 

1.82 

10.84 

1.86 

11 

12 

11.85 

1.88 

11.84 

1.93 

11.84 

1.98 

11.83 

2.03 

12 

13 

12.84 

2.03 

12.83 

2.09 

12.82 

2.15 

12.81 

2.20 

13 

14 

13.83 

2.19 

13.82 

2.25 

13.81 

2.31 

13.80 

2 37 

14 

15 

14.82 

2.35 

14.80 

2.41 

14.79 

2.48 

14.78 

2.54 

15 

16 

15.80 

2.5G 

15.79 

2.57 

15.78 

2.64 

15.77 

2.71 

16 

17 

16.79 

2.66 

16.78 

2.73 

16.77 

2.81 

16.75 

2.88 

17 

18 

17.78 

2.82 

17.77 

2.89 

17.75 

2.97 

17.74 

3.05 

18 

19 

18.77 

2.97 

18.75 

3.05 

18.74 

3. 14 

18.73 

3.22 

19 

20 

19.75 

3.13 

19.74 

3.21 

19.73 

3.30 

19.71 

3.39 

20 

21 

20.74 

3.29 

20.73 

3.38 

20.71 

3.47 

20.70 

3.56 

21 

22 

21.73 

3.44 

21.71 

3.54 

21.70 

3.63 

21.68 

3.73 

22 

23 

22.72 

3.60 

22.70 

3.70 

22.68 

3.80 

22.67 

3.90 

23 

24 

23.70 

3.75 

23.69 

3.86 

23.67 

3.96 

23.65 

4.06 

24 

25 

24.69 

3.91 

24.67 

4.02 

24.66 

4.13 

24.64 

4.23 

25 

26 

25.68 

4.07 

25.66 

4.18 

25.64 

4.29 

25.62 

4.40 

26 

27 

26.67 

4.22 

26.65 

4.34 

26.63 

4.46 

26.61 

4.57 

27 

28 

27.66 

4.38 

27.64 

4.50 

27.62 

4.62 

27.60  | 

4.74 

28 

29 

28 . 64 

4.54 

28.62 

4.66 

28.60 

4.79 

28.58 

4.91  i 

29 

30 

29.63 

4.69 

29.61 

4.82 

29.59 

4.95 

29.57 

5.08  | 

30 

31 

30.62 

4.85 

30.80 

4.98 

30.57 

5.12 

30.55 

572^ 

31 

32 

31.61 

5.01 

31.58 

5.14 

31.56 

5.28 

31.54 

5.42 

32 

33 

32.59 

5.16 

32.57 

5.30 

32.55 

5.45 

32.52 

5.59 

33 

34 

33.58 

5.32 

33.56 

5.47 

33.53 

5.61 

33.51 

5.76 

34 

35 

34.57 

5.48 

34.54 

5.63 

34.52 

5.78 

34.49 

5.93 

35 

36 

35.56 

5.63 

35.53 

5.79 

35.51 

5.94 

35.48 

6.10 

36 

37 

36.54 

5.79 

36.52 

5.95 

36.49 

6.11 

36.47 

6.27 

37 

38 

37.53 

5.94 

37.51 

6.11 

37.48 

6.27 

37.45 

6.44 

38 

39 

38.52 

6.10 

38.49 

6.27 

38.47 

6.44 

38.44 

6.60 

39 

40 

30.51 

6.26 

39.48 

6.43 

39.45 

6.60 

39.42 

6.77 

40 

41 

40.50 

6.41 

40.47 

6.59 

40.44 

6.77 

40.41 

6.94 

41 

42 

41  48 

6.57 

41.45 

6.75 

41.42 

6.92 

41.39 

7 11 

42 

43 

42  47 

6.73 

42.44 

6.91 

42.41 

7.10 

42.38 

1 i 28 

43 

44 

43  46 

6.88 

43.43 

7.07 

43.40 

7.26 

43.36 

7.45 

44 

15 

44.45 

7.04 

44.41 

7.23 

44.38 

7.43 

44.35 

7.62 

45 

46 

45.43 

7.20 

45.40 

7.39 

45.37 

7.59 

45.34 

, 7.79 

16 

47  , 

46.42 

7.35 

46.39 

7.55 

46.36 

7.76 

46.32 

7.96 

17 

48 

47  41 

7.51 

47.38 

7.72 

47.34 

7.92 

47.31 

8.13 

18 

49  ' 

48.40 

7.67 

48.36 

7.88 

48.33 

8.09 

48.29 

8.30 

49 

69  1 

49.38 

7.82 

49.35 

8.04 

49.32 

8.25 

49.28 

8.47 

50 

I| 

Dep. 

Lat. 

1 >ep. 

Lat. 

Dep. 

Lat. 

Dep. 

] Lat. 

: § 

a | 
Q 

R1  Deg. 

„ «0J  Dog.  | 

801 

Dog. 

m Deg. 

1 

TUAVKKSi;  I'AULK. 


91 


o 

6 

? 

9 Deg. 

93  Deg 

H Deg. 

9|  Deg 

O 

S' 

p 

g 

© 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

O 

© 

51 

50  37 

7.98 

50.34 

8.20 

50.30 

8.42 

50.26 

8.64 

51 

52 

51 .36 

8.13 

51.32 

8.36 

51.29 

8.58 

51.25 

8.81 

52 

63 

62.35 

8.29 

52.31 

8.52 

52.27 

8.75  j 

52.23 

8.98 

53 

54 

53.34 

8.45 

53.30 

8.68 

53.26 

8.91 

53.22 

9.14 

54 

65 

54.32 

8.60 

54  28 

8.84 

64.25 

9.08 

54.21 

9.31 

55 

56 

55.31 

8.76 

55.27 

9.00 

55.23 

9.24 

55.19 

9.48 

56 

67 

56.30 

8.92 

56.26 

9 16 

56.22 

9.41 

66.18 

9.65 

57 

58 

57.29 

9.07 

57.25 

9.32 

57.20 

9.57 

67.16 

9.82 

58 

59  I 68.27 

9.23 

58.23 

9 48 

58.19 

9.74 

58.15 

9. 99 

59 

60 

59.26 

9.39 

59.22 

9 fS4 

59.18 

9.90 

69.13 

10.16 

60 

61 

60.25 

9.54 

60.21 

9.81 

60.16 

10.07 

60.12 

10.33 

61 

62 

61.24 

9.70 

61.19 

9.97 

61.15 

10.23 

61.10 

10.50 

62 

63 

62.22 

9.86 

62.18 

10.13 

62.14 

10.40 

62.09 

10.67 

63 

64 

63.21 

10.01 

63.17 

10.29 

63.12 

10.56 

63.08 

10.84 

64 

65 

64.20 

10.17 

64.15 

10.45 

64.11 

10.73 

64.06 

11.01 

65 

66 

65.19 

10.32 

65.14 

10.61 

65.09 

10.89 

5.05 

11.18 

66 

67 

66.18 

10.48 

66.13 

10.77 

66.08 

11.06 

66.03 

11.35 

67 

68 

67.16 

10.64 

67.12 

10.93 

67.07 

11.22 

67.02 

11.52 

68 

69 

68.15 

10.79 

68.10 

11.09 

68.05 

11.39 

68.00 

11.69 

69 

70 

69.14 

10.95 

69.09 

11.25 

69.04 

11.55 

68.99 

11.85 

70 

71 

70.13 

11. li 

70.08 

11  41 

70.03 

11.72 

69.97 

12.02 

71 

72 

71.11 

11.26 

71.06 

11.57 

71.01 

11.88 

70.96 

12.19 

72 

73 

72.10 

11.42 

72.05 

11.73 

72.00 

12.05 

71.95 

12.36 

73 

74 

73.09 

11.58 

73.04 

11.89 

72.99 

12.21 

72.93 

12.53 

74 

75 

74.08 

11.73 

74.02 

12.06 

73.97 

12.38 

73.92 

12.70 

75 

76 

75.06 

11.89 

75.01 

12.22 

74.96 

12.64 

74.90 

12.87 

76 

77 

76.05 

12.05 

76.00 

12.38 

75.94 

12.71 

75.89 

13.04 

77 

78 

77.04 

12.20 

76.99 

12.54 

76.93 

12.87 

76.87 

13.21 

78 

79 

78.03 

12.36 

77.97 

12.70 

77.92 

13.04 

77.86 

13.38 

79 

80 

79.02 

12.51 

78.96 

12.86 

78.90 

13.20 

78.84 

13.55 

80 

81 

80.00 

12.67 

79.95 

13.02 

79.89 

13.37 

79.83 

13.72 

81 

82 

80.99 

12.83 

80.93 

13.18 

80.88 

13.53 

80.82 

13.89 

82 

83 

81.98 

12.98 

81.92 

13.34 

81.86 

13.70 

81.80 

14.06 

83 

84 

82.97 

13.14 

82.91 

13.50 

82.85 

13.86 

82.79 

14.23 

84 

85 

83.95 

13.30 

83.89 

13.66 

83.83 

14.03 

83.77 

14.39 

85 

86 

84.94 

13.45 

84.88 

13.82 

84.82 

14.19 

84.76 

14.56 

86 

87 

85.93 

13.61 

85.87 

13.98 

85.81 

14.36 

85.74 

14.73 

87 

88 

86.92 

13.77 

86.86 

14.15 

86.79 

14.62 

86.73 

14.90 

88 

89 

87.90 

13.92 

87.84 

14.31 

87.78 

14.69 

87.71 

15.07 

89 

90 

88.89 

14.08 

88.83 

14.47 

88.77 

14.85 

88.70 

15.24 

90 

91 

89.88 

14.24 

89.82 

14.63 

89.75 

15.02 

89.69 

15.41 

91 

92 

90.87 

1 4.39 

90.80 

14.79 

90.74 

15.18 

90  67 

15.68 

92 

93 

91.81 

14.55 

91.79 

14.95 

91.72 

15.35 

91.66 

16.76 

93 

94 

92.84 

14.70 

92.78 

15.11 

92.71 

16.51 

92.64 

15.92 

94 

95 

93.83 

14.86 

93.76 

15.27 

93.70 

16.68 

93.63 

16.09 

95 

96 

94.82 

15.02 

94.75 

15.43 

94.68 

15.84 

94.61 

16.26 

96 

97 

95.81 

15.17 

95.74 

15.59 

95.67 

16.01 

95.60 

16  43 

97 

98  i 

96.79 

15.33 

96.73 

15.75 

96.66 

16.17 

96.58 

16.60 

98 

99 

97.78 

15.49 

97.71 

15.91 

97.64 

16.34 

97.57 

16.77 

99 

100 

98.77 

15.64 

J98.70 

16.07 

98.63 

16.50 

98.56 

16.93 

100 

© 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

I at. 

® 

0 

c 

Q 

81  Deg. 

| 80f  Deg. 

80f  Deg 

80$  Dog. 

od 

CD 

a 

92 


TRAVERSE  TABLE 


q 

w’ 

*-♦ 

P 

10  Deg. 

10?  Deg. 

I0i 

Deg. 

»0|  Dog. 

q 

3 

O 

® 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

0 

9 

1 

0.98 

0.17 

0.98 

0.18 

0.98 

0.  18 

0.98 

0T§ 

1 

2 

1.97 

0.35 

1.97 

0.36 

1.97 

0.36 

1.96 

0.37 

2 

3 

2.95 

0.52 

2.95 

0.53 

2.95 

0.55 

2.95 

0 66 

3 

4 

3.94 

0.69 

3.94 

0.71 

3.93 

0.73 

3.93 

0.75 

4 

5 

4.92 

0.87 

4.92 

0.89 

4.92 

0.91 

4.91 

0 93 

5 

6 

5.91 

1 .04 

5.90 

1.07 

5.90 

1.09 

5.89 

1 12 

S 

7 

6.89 

1.22 

6.89 

1.25 

6.88 

1 28 

5.88 

1 31 

7 

8 

7.88 

1.39 

7.87 

1.42 

7.87 

1 46 

7.86 

1.49 

8 

9 

8.86 

1.56 

8.86 

1.60 

8.85 

1.64 

8.84 

1.68 

0 

10 

9.85 

1.74 

9.84 

1.78 

9.83 

1.82 

9.82 

1.87 

10 

11 

10  83 

I .91 

10.82 

1.96 

10782 

' 2.00 

10.81 

" 2 .05 

11 

12 

11  82 

2.08 

11.81 

2.14 

11.80 

2.19 

11.79 

2.24 

12 

13 

12.80 

2.26 

12.79 

2.31 

12.78 

2.37 

12.77 

2.42 

13 

14 

13.79 

2.43 

13.78 

2.49 

13.77 

2.55 

13.75 

2.61 

14 

15 

14.77 

2.60 

14.76 

2.67 

14.75 

2.73 

14.74 

2.80 

16 

16 

15.76 

2.78 

15.74 

2.85 

15.73 

2.92 

15.72 

2.98 

16 

17 

16.74 

2.95 

16.73 

3.03 

16.72 

3.10 

16.70 

3. 17 

17 

18 

17.73 

3.13 

17.71 

3.20 

17.70 

3.28 

17.68 

3.36 

18 

19 

18.71 

3.30 

18.70 

3.38 

18.68 

3.46 

18.67 

3.54 

19 

20 

19.70 

3.47 

19.68 

3.56 

19.67 

3.64 

19.65 

3.73 

20 

21 

20 . 68 

3.65 

20.66 

3.74 

20.65 

3.83 

20.63 

3.92 

21 

22 

21.67 

3.82 

21.65 

3.91 

21.63 

4.01 

21.61 

4.10 

22 

23 

22.65 

3.99 

22.63 

4.09 

22.61 

4.19 

22.60 

4.29 

23 

24 

23.64 

4.17 

23.62 

4.27 

23.60 

4.37 

23 . 58 

4.48 

24 

25 

24.62 

4,34 

24.60 

4.45 

24.58 

4.56 

24.56 

4.66 

25 

26 

25.61 

4.51 

25.59 

4.63 

25.56 

4.74 

25.54 

4.85 

26 

27 

26.59 

4.69 

26.57 

4.80 

26.55 

4.92 

26.53 

5.04 

27 

28 

27.57 

4.86 

27.55 

4.98 

27.53 

5.10 

27.51 

5.22 

28 

29 

28.56 

5.04 

28.54 

5.16 

28.51 

5.28 

28.49 

5.41 

29 

30 

29.54 

5.21 

29.52 

5.34 

29.50 

5.47 

29.47 

5.60 

30 

31 

30 .53 

5.38 

30.51 

5.52 

80.48 

5.65 

30.46 

■5778 

31 

32 

31.51 

5.56 

31.49 

5.69 

31.46 

5.83 

31.44 

5.97 

32 

33 

32.50 

5.73 

32.47 

5.87 

32.45 

6.01 

32.42 

6.16 

33 

34 

33.48 

5.90 

33.46 

6.05 

33.43 

6.20 

33.40 

6.34 

34 

35 

34  4~ 

6.08 

34.44 

6.23 

34.41 

6.38 

34.39 

6.53 

35 

36 

35.45 

6.25 

35.43 

6.41 

35.40 

6.56 

35.37 

6.71 

36 

37 

36.44 

6.42 

36.41 

6.58 

36.38 

6.74 

36.35 

6.90 

37 

38 

37.42 

6.60 

37.39 

6.76 

37.36 

6.92 

37.33 

7.09 

38 

39 

38.41 

6.77 

38.38 

6.94 

38.35 

7.11 

38.32 

7.27 

39 

40 

39.39 

6.95 

39.36 

7. 12 

39.33 

7.29 

39.30 

7.46 

40 

41 

40.38 

7.12 

40.35 

7.30 

40.31 

7.47 

40.28 

7.65 

41 

42 

141.36 

7.29 

41.33 

7.47 

41.30 

7.65 

41.26 

7.83 

42 

43 

42.35 

7.47 

42.31 

7.65 

42.28 

7.84 

42.25 

S.02 

43 

44 

43.33 

7.64 

43.30 

7.83 

43.26 

8.02 

43.23 

8.21 

44 

45 

144.32 

7.81 

44.28 

8.01 

44.25 

8.20 

44.21 

8.39 

15 

46 

45.30 

! 7.99 

| 45.27 

8.19 

45.23 

8.38 

45.19 

8.68 

46 

47 

46  29 

1 8.16 

46.25 

8.36 

46.21 

8.57 

46.18 

8.77 

47 

48 

: 47.27 

8.34 

47.23 

8.54 

47.20 

8.75 

47.16 

8.95 

48 

49 

48.26 

8.51 

! 48.22 

8.72 

48.18 

8.93 

48.14 

9.14 

49 

oO 

1 49.24 

8.68 

49.20 

8.90 

49.16 

9.11 

49.12 

9.33 

50 

i 

| Dep. 

Lat. 

1 Dep. 

L^t. 

Dep. 

Lat. 

Dep. 

Lat. 

s 

q 

eg 

b 

80  Deg 

1 . 

79.?  Deg. 

79J  Deg. 

79?  Deg. 

3 

1 2 
Li 

TRAVERSE  TaULE. 


9<j 


l> 

10  Deg. 

101  Deg. 

10*  Deg. 

10|  Deg.  j 

O 

s 

p 

* 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

o 

a 

i 51 

50.23 

8.86 

50. 19 

9.08 

50.15 

9.29 

50"70 

“9.51  [ 

51 

52 

51.21 

9.03 

51.17 

9.25 

51.13 

9.48 

51 .09 

9 70  1 

52 

’ 53  52 . 1 9 

9.20 

52.15 

9.43 

52.11 

9.66 

52.07 

9.89 

53 

54 

53.18 

9.38 

53.14 

9.61 

53.10 

9.84 

53.05 

10.07 

54 

55 

54.16 

9.55 

54.  12 

9.79 

54.08 

10.02 

54.03 

10.26 

55 

56 

55.15 

9.72 

55.11 

9 96 

55.06 

10.21 

55.02 

10.45 

56 

57 

56.13 

9.90 

56.09 

10. 14 

56.05 

10.39 

56.00 

10.63 

57 

58 

57.12 

10.07 

57.07 

10.32 

57.03 

10.57 

56.98 

10.82 

58 

59 

58.10 

10.25 

58.06 

10.50 

58.01 

10.75 

57.96 

U.00 

59 

60 

59.09 

10.42 

59.04 

10.68 

59.00 

10.93 

58.95 

11.19 

60 

61 

60.07 

10.59 

60.03 

10.85 

59.98 

11.12 

59.93 

11.38 

'61 

62 

61.06 

10.77  ! 

61.01 

11.03 

60.96 

11.30 

60.91 

11.56 

62 

63 

62.04 

10.94 

61  .99 

1 1 .21 

61.95 

11.48 

61.89 

11.75 

63 

64 

63.03 

11.11 

62.98 

11.39 

62.93 

11.66 

62.88 

11.94 

64 

65 

64.01 

11.29 

63.96 

11.57 

63.91 

11.85 

63.86 

12. 12 

65 

66 

65.00 

11.46 

64.95 

11.74 

64.89 

12.03 

64.84 

12.31 

66 

67 

65.98 

11.63 

65.93 

11.92 

65.88 

12.21 

65.82 

12.50 

67 

68 

66.97 

11.81 

66.91 

12.10 

66.86 

12.39 

66.81 

12.68 

68 

69 

67.95 

11.98 

67.90 

12.28 

67.84 

12.57 

67.79 

12.87 

69 

70 

68.94 

12.16 

68.88 

12.46 

68.83 

12.76 

68.77 

13.06 

70 

71 

69.92 

12.33 

89.87 

12.63 

69.81 

12.94 

69.75 

13.24 

71 

72 

70.91 

12.50 

70.85 

12.81 

70.79 

13.12 

70.74 

13.43 

72 

73 

71.89 

12.68 

71.83 

12.99 

71.78 

13.30 

71.72 

13.62 

73 

74 

72.88 

12.85 

72.82 

13.17 

72.76 

13.49 

72.70 

13.80 

74 

75 

73.86  | 

13.02 

73.80 

13.35 

73.74 

13.67 

73.68 

13.99 

75 

76 

74.85  l 

13.20 

74.79 

13.52 

74.73 

13.85 

74 . 67  i 

14.18 

76 

77 

75.83 

13.37 

75.77 

13.70 

75.71 

14.03 

75.65 

14.36  1 

l 77 

78 

76.82 

13.54 

76.76 

13.88 

76.69 

14.21 

76 . 63 

14.55 

78 

79 

77.80 

13.72 

77.74 

14.06 

77.68 

14.40 

77.61 

14.74 

79 

80 

78.78 

| 13.89 

78.72 

14.24 

78.66 

14.58 

78.60 

14.92 

80 

81 

79.77 

i 14.07 

79.71 

14.41" 

79.64 

14.76 

79.58 

15.11 

81 

82 

80 . 75 

14.24 

80.69 

14.59 

80.63 

14.94 

80.56 

15.29 

82 

83 

81.74 

1 14.41 

81.68 

14.77 

81. Cl 

15.13 

81  .54 

15.48 

83 

84 

82.72 

1 14.59 

82.66 

; 14.95 

82.59 

15.31 

82.53 

15.67 

84 

85 

83.71 

j 14.76 

83.64 

. 15.13 

83.58 

15.49 

83.51 

15.85 

85 

86 

84.  G9 

| 14.93 

84.63 

15.30 

84.56 

15.67 

84.49 

16.04 

86 

87 

85.68 

! 15.11 

85.61 

, 15.48 

85.54 

15.85 

85.47 

16.23 

87 

88 

86.66 

: 15.28 

86.60 

; 15.66 

86.53 

16.04 

86.46 

16.41 

| 88 

89 

87.65 

j 15.45 

87.58 

1 15.84 

87.51 

16.22 

87.44 

i 16.60 

89 

90 

88.63 

15.63 

88.56 

| 16.01 

88.49 

16.40 

88.42 

! 16.79 

! 90 

91 

89.62 

15.80 

89.55 

i 16.19 

89.48 

16.58 

89.40 

"16.97 

91 

92 

90.60 

15.98 

90.53 

16.37 

90.46 

16.77 

90.39 

17.16 

92 

93  91.59 

16.15 

91.52 

16.55 

91.44 

16.95 

91.37 

17.35 

93 

94 

i 92.57 

16.32 

92.50 

16.73 

92.43 

17.13 

92.35 

17.53 

34 

95  93.56 

16.50 

93.48 

16.90 

93.41 

17.31 

93.33 

17.72 

95 

96 

94 . 54 

16.67 

94.47 

17.08 

94.39 

17.49 

94.32 

17.91 

96 

97 

95  53 

16.84 

95.45 

17.26 

95.38 

17.68 

95.30 

18.09 

97 

98 

96 . 5 1 

17.02 

96.44 

17.44 

96.36 

17.86 

96.28 

18.28 

98 

99 

97 . 50 

17.19 

97.42 

17.62 

97.34 

18.04 

97.26 

18.47 

99 

100 

98.48 

17.36 

98.40 

17.79 

98^33 

18.22 

98.25 

18.65 

00 

0) 

o 

C 

Dep. 

Lat. 

Dep. 

i Lat. 

i 

Dep. 

Lat. 

Dep. 

Lat. 

a 3 

g 

t) 

to 

80  Dog. 

' 791  Deg. 

79* 

Deg. 

79*  Deg. 

ad 

s 

1 b 

1 

94 


1R  AVERSE  TABLE 


1 

o 

i 

11  Deg. 

1U  Deg. 

1 

11* 

Deg. 

11}  Deg. 

O 

3 

$ 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  1 

Dep 

3 

8 

2 

\ 

5 

€ 

7 

8 

9 

10 

0.98 
1.96 
2.94 
3.93 
4.91 
5.89 
6.87 
? .85 
8.83 
9.82 

0.19 

0.38 

0.57 

0.76 

0.95 

1.14 

1.34 

1.53 

1.72 

1.91 

0.98 

1.96 

2.94 

3.92 

4.90 

5.88 

6.87 

7.85 

8.83 

9.81 

0.20 

0.39 

0.59 

0.78 

0.98 

1.17 

1 .37 

1 .56 
1.76 
1.95 

(f.  98 
1.96 
2.94 
3.92 

4 90 

5 88 
6.86 
7.84 
8.82 
9.80 

0.20 

0.40 

0.60 

0.80 

1 .00 
i 2 0 

1 .4\J 
1.59 
1.79 
1.99 

| 0.98 
: 1 .96 

i 2.94 
3.92 
4.90  , 
5.87  ! 
6.85  j 
7.83 
8.81 
9.79 

0.20 

0.41 

0.61 

0.82 

1.02 

1.22 

1.43 

1.63 

1.83 

2.04 

l 

4 

3 

4 

5 

8 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

10.80 

11.78 

12.76 

13.74 

14.72 

15.71 

16.69 

17.67 

18.65 

19.63 

2.10 

2.29 

2.48 

2 67 

2 86 
3.05 
3.21 
3.41 
3.63 
3.82 

10.79 

11.77 

12.75 

13.73 

14.71 

15.69 

16.67 

17.65 

18.63 

19.62 

2.15 

2.34 

2.54 

2.73 

2.93 

3.12 

3.32 

3.51 

3.71 

3.90 

10.78 

11.76 

12.74 

13.72 

14.70 

15.68 

16.66 

17.64 

18.62 

19.60 

2.19 

2.39 

2.59 

2.79 

2.99 

3.19 

3.39 

3.59 

3.79 

3.99 

10.77 

11.75 

12.73 

13.71 

14.69 

15.66 

16.64 

17.62 

18.60 

19.58 

2.24 

2.44 

2.65 
2.85 

3.06 
3.26 
3.46 

3.66 
3.87 

4.07 

‘ 'll 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

20.61 

4.01 

20.60 

4.10 

20.58 

4.19 

20.56 

4.28 

21 

22 

21.60 

4.20 

21.58 

4.29 

21.56 

4.39 

21.54 

4.48 

22 

23 

22.58 

4.39 

22.56 

4.49 

22.54 

4.59 

1 22.52 

4.68 

23 

24 

23.56 

4.58 

23.54 

4.68 

23.52 

4.78  ! 

23.50 

4.89 

24 

25 

24.54 

4.77 

24.52 

4.88 

24.50 

4.98 

24.48 

5.09 

25 

26 

25.52 

4.96 

25.50 

5.07 

25.48 

5.18 

25.46 

5.30 

26 

27 

26.50 

5.15 

26.48 

5.27 

26.46 

5.38 

26.43 

5.50 

27 

28 

27.49 

5.34 

27.46 

5.46 

27.44 

5.58 

27.41 

5.70 

28 

29 

28.47 

5.53 

28.44 

5.66 

28.42 

5.78 

28.39 

5.91 

29 

30 

29.45 

5.72 

29.42 

5.85 

29.40 

5.98 

29.37 

6.11 

30 

31 

30.43 

5.92 

30.40 

6.05 

30.38 

6.18 

30.35 

6.31 

31 

32 

31.41 

6.11 

31.39 

6.24 

31.36 

6.38 

31.33 

6.52 

32 

33 

32.39 

6 30 

32.37 

6.44 

32.34 

6.58 

32.31 

6.72 

33 

34 

33.38 

6.49 

33.35 

6.63 

33.32 

6.78 

33.29 

6.92 

34 

35 

34.36 

6.68 

34.33 

6.83 

34.30 

! 6.98 

34.27 

7.  13 

35 

36 

35.34 

6.87 

35.31 

7.02 

35.28 

7. 18 

35.25 

7 33 

36 

37 

36.32 

7.06 

36.29 

7.22 

36.26 

7.38 

36.22 

7. 53 

37 

38 

37.30 

7.25 

37.27 

7.41 

37.24 

7.58 

37.20 

7.74 

38 

39 

38.28 

7.41 

38.25 

7.61 

38.22 

7.78 

38.18 

7.94 

39 

40 

39.27 

7.63 

39.23 

7.80 

39.20 

7.97 

39.16 

8.15 

40 

41 

40.25 

7.82 

40.21 

! 8.00 

40.18 

8.17 

40.14 

8.35 

41 

42 

11.23 

8.01 

41.19 

8. 19 

41.16 

8.37 

41.12 

8.55 

45 

43 

, 42.21 

8.20 

42.17 

8.39 

42.14 

1 8.57 

42.10 

1 8.76 

43 

11  ; 43.19 

8.40 

43.15 

8.58 

43.12 

i 8.77 

43.08 

8.96 

41 

45  ! 44  17 

! 8.59 

44.14 

8.78 

44.10 

i 8.97 

44.06 

9.16 

45 

‘ 46 

45.15 

8.78 

45.12 

8.97 

45.08 

9.17 

45.04 

9.37 

46 

47 

i 46.14 

8.97 

46.10 

9.17 

46.06 

9.37 

46.02 

9.57 

47 

48 

! 47.12 

9.16 

47.08 

9.36 

47.04 

9.57 

46.99 

9.78 

48 

49 

1 48. 10 

9.35 

48.06 

9.50 

48.02 

9.77 

47.97 

9.98 

49 

50 

, 49.08 

| 9.54 

49.04 

| 9J^5. 

49.00 

9.97 

48.95 

10.18 

50 

g 

g 

1 Dep. 

j Lat. 

Dep. 

{ Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

9 

5 

\ 9 

1 a 

( 

79  Deg 

1 

78}  Deg. 

8$  Deg. 

78}  Deg. 

! -a 

* 

TKAVEltSK  TABLE. 


95 


© 

S" 

11 

Deg. 

lli  Dey. 

H£ 

Deg. 

11|  Deg. 

O 

& 

3 

O 

IS 

Lat.  1 Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| Dep. 

b 

o 

® 

51 

50.06 

9.73 

60.02 

9.95 

49.98 

10.17 

49.93 

| 10.39 

51 

52 

51.04 

9.92 

51.00 

10.14 

50.96 

10.37 

50.91 

1 10.59 

52 

53 

52.03 

10.11 

f>l  98 

10.34 

51.94 

10.57 

51.89 

10.79 

53 

54 

53.01 

10.30 

52.96 

10.53 

52.92 

10.77 

52.87 

11.00 

54 

55 

53.99 

! 10.49 

53.94 

10.73 

53.90 

10.97 

53.85 

11.20 

65 

56 

54.97 

10.69 

54.92 

10.93 

54.88 

11.16 

54.83 

11.40 

56 

57 

55.95 

10.88 

55.90 

11.12 

55.86 

11.36 

55.81 

11.61 

5? 

58 

56.93 

11.07 

56.89 

11.32 

56.84 

11.56 

56.78 

11.81 

58 

59 

57.92 

11.26 

57.87 

11.51 

57.82 

11.76 

57.76 

12.01 

59 

60 

58.90 

11.45 

58.85 

11.71 

58.80 

11.96 

58.74 

12.22 

60 

61 

59.88 

11.64 

59.83 

11.90 

59.78 

12.16 

59.72 

12.42 

61 

62 

60.86 

11.83 

60.81 

12.10 

60.76 

12.36 

60.70 

12.63 

62 

63 

61.84 

I 12.02 

61.79 

12.29 

61.74 

12.56 

61.68 

12.83 

63 

64 

62.82 

12.21 

62.77 

12.49 

62.72 

12.76 

62.66 

13.03 

64 

65 

63.81 

1 12.40 

63.75 

12.68 

63.70 

12.96 

63.64 

13.24 

65 

66 

64.79 

1 12.59 

64.73 

12.88 

64.68 

13.16 

64.62 

13.44 

66 

67 

65.77 

12.78 

65.71 

13.07 

65.66 

13. J6 

65.60 

13.64 

67 

68 

66.75 

12.98 

66.69 

13.27 

66.63 

13.56 

66.58 

13.85 

68 

69 

67.73 

13.17 

67.67 

13.46 

67.61 

13.76 

67.55 

14.05 

69 

70 

68.71 

13.36 

68.66 

13.66 

68.59 

13.96 

68.53 

14.25 

70 

71 

69.70 

13.55 

69.64 

13.85 

69.57 

14.16 

69.51 

14.46 

71 

72 

70.68 

13.74 

70.62 

14.05 

70.55 

14.35 

70.49 

14.66 

72 

73 

71.66 

13.93 

71.60 

14.24 

71.53 

.'4.55 

71.47 

14.87 

73 

74 

72.64 

| 14.12 

72.58 

14.44 

72.51 

14.75 

72.45 

15.07 

74 

75 

73.62 

i 14.31 

73.56 

14.63 

73.49 

14.95 

73.43 

15.27 

75 

76 

74.60 

! 14.50 

74.54 

14.83 

74.47 

15.15 

74.41 

15.48 

76 

77 

75.59 

14.69 

75.52 

15.02 

75.45 

15.35 

75  39 

15.68 

77 

78 

76.57 

14.88 

76.50 

15.22 

76.43 

15.55 

76.37 

15.88 

78 

79 

77.55 

15.07 

77.48 

15.41 

77.41 

15.75 

77.34 

16.09 

79 

80 

78.53 

15.26 

78.46 

15.61 

78.39 

15.95 

78.32 

16.29 

80 

'81 

79.51 

15.46 

79.44 

15.80 

79.37 

1 16.15 

79.30 

16.49 

81 

82 

80.49 

15.65 

80.42 

16.00 

80.35 

16.35 

80.28 

16.70 

82 

83 

81.48 

15.84 

81.41 

16.19 

81.33 

16.55 

81.26 

16.90 

83 

84 

82.46 

16.03 

82.39 

16.39 

82.31 

16.75 

82.24 

17.11 

84 

85 

83.44 

16.22 

183.37 

16.58 

83.29 

16.95 

83.22 

17.31 

85 

86 

84.42 

16.41 

84.35 

16.78 

84.27 

17.15 

84.20 

17.51 

86 

87 

85.40 

16.60 

85.33 

16.97 

85.25 

17.35 

85.18 

17.72 

87 

88 

86.38 

16.79 

86.31 

17.17 

86.23 

17.54 

86.16 

17.92 

88 

89 

87.36 

16.98 

87.29 

17.36 

87.21 

17.74 

87.14 

18.12 

89 

90 

88.35 

17.17 

88.27 

17.56 

88.19 

17.94 

88.11 

18.33 

90 

91 

89.33 

17.36 

89.25 

17.75 

89.17 

18.14 

89.09 

18.53 

91 

92 

90.31 

17.55 

90.23 

17.95 

90.15 

18.34 

90.07 

18.74 

92 

93  | 

91.29 

17.75 

91.21 

18.14 

91.13 

18.54 

91.05 

18.94 

93 

94 

92.27  i 

17.94 

92.19 

18.34 

92.11  1 

18.74 

92.03 

19.14 

94 

95 

93.25 

18.13 

93.17 

18.53 

93.09 

18.94 

93.01 

19.35 

95 

96 

94.24  | 

18.32 

94.16 

18.73 

94.07 

.9.14 

93.99 

19.55 

96 

37  1 

95.22 

18.51 

95.14 

18.92 

95.05 

19.34 

94.97 

19.75 

97 

98 

96.20  1 

18.70 

96.12 

19.12 

96.03 

19.54 

95.95 

19.96 

98 

99 

97.18 

18.89 

97.10 

19.31 

97.01 

19.74 

96.93 

20.16 

99 

100 

98.16 

19.08 

98.08 

19.51 

97.99 

19.94 

97.90 

20.36 

100 

| 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

S 

1 1 
Q | 

79  Deg. 

78|  Deg. 

7*4  Deg. 

m Deg. 

3 

* 

96 


Til  AVERSE  TAIJLE 


0 

1 
a 

12  Deg 

124  Deg. 

Deg. 

I2i  Deg. 

1 

fl 

Lat. 

Dep. 

Lat. 

Dep. 

Lat  ! 

Dep. 

Lat.  j 

Dep. 

1 

i 

0.98 

0.21 

0.98 

0.21 

0.98 

(K  22 

~079F| 

0.22 

1 

a 

1.96 

0.42 

1.95 

0.42 

1 . 95 

0.43 

1.95  1 

0.44  1 

2 

3 

2.93 

0.62 

2.93 

0.64 

2.93 

0.65 

2.93 : 

0.66  ; 

3 

4 

3.91 

0.83 

3.91 

0.85 

3.91 

0.87 

3.90  1 

0.88  ; 

4 

5 

4.89 

1.04 

4.89 

1.06 

4.88 

1.08 

4.88 

1.10  | 

5 

6 

5.87 

1.25 

5.86 

1.27 

5.86  | 

1.30 

5.85 

1 .32 

9 

7 

6.85 

1.46 

6.84 

1.49 

6 83  1 

1 .52 

6.83  i 

1 54 

7 

8 

7.83 

1.66 

7.82 

1.70 

7.81 

1.73 

7.80 

1 77 

8 

9 

8.80 

1.87 

8.80 

1.91 

8.79 

1.95 

8.78 

1.99 

9 

10 

9.78 

2.08 

9.77 

2.12 

9 . 76 

2.16 

9.75 

2.21 

10 

11 

10.76 

2.29 

10.75 

2.33 

10.74  ; 

~ 2.38 

10.73 

2.43 

11 

12 

11.74 

2.49 

11.73 

2.55 

11.72 

2.60 

11.70 

2.65 

12 

13 

12.72 

2.70 

12.70 

2.76 

12.69 

2.81 

12.68 

2.87 

13 

14 

13.69 

2.91 

13.68 

2.97 

13.67 

3.03 

13.65 

3.09 

14 

15 

14.67 

3.12 

14.66 

3.  18 

14.64 

3.25 

14.63 

3.31 

15 

16 

15.65 

3.33 

15.64 

3.39 

15.62 

3.46 

15.61  1 

3.53 

16 

17 

16.63 

3.53 

16.61 

3.61 

16.60 

3.68 

16.58 

3.75 

17 

18 

17.61 

3.74 

17.59 

3.82 

17.57 

3.90 

17.56 

3.97 

! 18 

19 

18.58 

3.95 

18.57 

4.03 

18.55 

4.11 

18.53 

4. 19 

19 

20 

19.56 

4.16 

19.54 

4.24 

19.53 

4.33 

19.51 

4.41 

20 

21 

20.54 

4.37 

20.52 

4.46 

20.50 

4.55 

20.48 

4.63 

21 

22 

21.52 

4.57 

21.50 

4.67 

21.48 

4.76 

21.46 

4.86 

22 

23 

22.50 

4.78 

22.48 

4.88 

22.45 

4.98 

22.43 

5.08 

23 

24 

23.48 

4.99 

23.45 

5.09 

23.43 

5.19 

23.41 

5.30 

24 

25 

24.45 

5.20 

24.43 

5.30 

24.41 

5.41 

24.38 

5.52 

25 

26 

25.43 

5.41 

25.41 

5.52 

25.38 

5.63 

25.36 

5.74 

26 

27 

26.41 

5.61 

26.39 

5.73 

26.36 

5.84 

26.33 

5.96 

27 

28 

27.39 

5.82 

27.36 

6.94 

27.34 

6.06 

27.31 

6.18 

28 

29 

28.37 

6.03 

28.34 

6.15 

28.31 

6.28 

28.28 

6.40 

29 

30 

29.34 

6.24 

29.32 

6.37 

29.29 

6.49 

29.26 

i 6.62 

30 

31 

1 30.32 

6.45 

30.29 

6.58 

30.27 

6.71 

30.24 

6.84 

31 

32 

31.30 

6.65 

31.27 

6.79 

31.24 

6.93 

31.21 

! 7.06 

32 

33 

i 32.28 

6.86 

32.25 

7.00 

32.22 

7.14 

32.19 

7.28 

33 

34 

33.26 

7.07 

33.23 

7.21 

33.19 

7.36 

33.16 

7.50 

34 

35 

34.24 

7.28 

34.20 

7.43 

34.17 

7.58 

34.14 

7.72 

35 

36 

35.21 

7.48 

35.18 

7.64 

35.15 

7.79 

35.11 

7.95 

36 

37 

36.19 

7.69 

36.16 

7.85 

36.12 

8.01 

36.09 

8.17 

37 

38 

37.17 

7.90 

37.13 

8.06 

37.10 

8.22 

37.06 

8.39 

38 

39 

38.15 

8.11 

38.11 

8.27 

38.08 

8.44 

38.04 

8.61 

! 39 

40 

39.13 

8.32 

39.09 

8.49 

39.05 

8.66 

39.01 

j 8.83 

1 40 

41 

40.10 

8.52 

40.07 

8.701 

40.03 

8.87 

39.99 

f 9 05 

'll 

42 

41.08 

8.73 

41.04 

8.91 

41.00 

9.09 

40.96 

9.27  1 4* 

43 

' 42.06 

8.94 

42.02 

9.12 

41.98 

9.3i 

41.94 

1 4.49  i 43 

44 

, 43.04 

9.15 

43.00 

9.34 

42.96 

9.52 

42.92 

l 9.71 

44 

45 

| 44.02 

9.30 

43.98 

9.55 

43.93 

9.74 

43  89 

9.93 

4* 

46 

44.99 

9.56 

44.95 

9.76 

44.91 

9.96 

1 44.87 

, 10.15  ! 4fl 

47 

45.97 

9.77 

45.93 

9.97 

45.89 

10.17 

45. 8-k 

| i0.37 

! 47 

48 

46.95 

9.98 

46.91 

10.18 

48.86 

10.39 

' 46.82 

’0  59 

j 48 

4G 

47.93 

10.19 

47.88 

10  40 

47.84 

10.61 

47.79 

10  81 

1 49 

5C 

48.91 

10.40 

48.86 

io.  ei 

48.81 

10.82 

48.77 

11.03 

1 52 

T 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 © 

1 o 

' fl 

Q 

78  Deg. 

77|  Deg. 

77 1 

* ’ I 

Deg. 

77}  Dog. 

$ 

tn 

1 o 

traverse  table. 


97 


2 

5* 

12  Dag. 

12\  Deg. 

12* 

Dog. 

12|  Deg. 

1 

D 

or 

pc 

3 

a 

? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| Dep. 

Lat. 

Dep. 

O 

? 

5: 

49  89 

10.60 

49.84 

10.82 

49.79 

| 11.04 

49.74 

11.26 

51 

52 

50.86 

10.81 

50.82 

11.03 

50.77 

11.25 

50.72 

11.48 

62 

53 

51.84 

11.02 

51.79 

11.25 

51.74 

11.47 

51.69 

11.70 

53 

54 

52.82 

11.23 

52.77 

11.46 

52.72 

11.69 

52.67 

11.92 

54 

si 

53.80 

11.44 

53.75 

11.67 

53  70 

11.90 

53.64 

12.14 

55 

56 

54.78 

11.64 

64.72 

11.88 

54.67 

I 12.12 

54.62 

12.36 

56 

57 

55.75 

11.85 

55.70 

12.09 

55.65 

12.34 

55.59 

12.58 

57 

58 

56.73 

12.06 

56.68 

12.31 

56.63 

1 12.55 

56.57 

12.80 

58 

59 

57.71 

12.27 

57.66 

12.52 

57.60 

12.77 

57.55 

13.02 

59 

60 

58.69 

12.47 

58.63 

12.73 

58.58 

12.99 

58.52 

13.24 

60 

61 

59.67 

12.68 

59.61 

12.94 

59.55 

13.20 

59.50 

13.46 

| 61 

62 

60.65 

12.89 

60.59 

13.16 

60.53 

13.42 

60.47 

13.68 

62 

63 

61.62 

13.10 

61.57 

13.37 

61.51 

13.64 

61.45 

13.90 

63 

64 

62.60 

13.31 

62.54 

13.58 

62.48 

13.85 

62.42 

14.12 

64 

65 

63.58 

13.51 

63.52 

13.79 

63.46 

14.07 

63.40 

1435 

65 

66 

64.56 

13.72 

64.50 

14.00 

64.44 

14.29 

64.37 

14.57 

66 

67 

65.54 

13.93 

65.47 

14.22 

65.41 

14.50 

65.35 

14.79 

67 

68 

66.51 

14.14 

66.45 

14.43 

66.39 

14.72 

66.32 

15.01 

68 

69 

67.49 

14.35 

67.43 

14.64 

67.36 

14.93 

67.30 

15.23 

69 

70 

68.47 

14.55 

68.41 

14.85 

68.34 

15.15 

68.27 

15.45 

70 

71 

09.45 

14.76 

69.38 

15.06 

69.32 

15.37 

69.25 

15.67 

71 

72 

70.43 

14.97 

70.36 

15.28 

70.29 

15.58 

70.22 

15.89 

72 

73 

71.40 

15.18 

71.34 

15.49 

71.27 

15.80 

71.20 

16.11 

73 

74 

72.38 

15.39 

72.32 

15.70 

72.25 

16.02 

72.18 

16.33 

74 

75 

73.36 

15.59 

73.29 

15.91 

73.22 

16.23 

73.15 

16.55 

75 

76 

74.34 

15.80 

74.27 

16.13 

74.20 

16.45 

74.13 

16.77 

76 

77 

75.32 

16.01 

75.25 

16.34 

75.17 

16.67 

75.10 

16.99 

77 

78 

76.30 

16.22 

76.22 

16.55 

76.15 

16.88 

76.08 

17.21 

78 

79 

77.27 

16.43 

77.20 

16.76 

77.13 

17.10 

77.05 

17.44 

79 

80 

78.25 

16.63 

78.18 

16.97 

78.10 

17.32 

78.03 

17.66 

80 

'81 

79.23 

16.84 

79.16 

17.19 

79.08 

17.53 

79.00 

17.88 

81 

82 

90.21 

17.05 

80.13 

17.40 

80.06 

17.75 

79.98 

18.10 

82 

83 

81.19 

17.26 

8 i . 1 1 | 

17.61 

81.03 

17.96 

80.95 

18.32 

83 

84 

82.16 

17.46 

82.09  | 

17.82 

82.01 

18.18 

81.93 

18.54 

84 

85 

63.14 

17.67 

83.06 

IS  04 

82.99 

18.40 

82.90 

18.76 

85 

86 

84. 12 

17.88 

84.04 

18.25 

83.96 

18.61 

83.88 

18.98 

86 

87 

85.10 

18.09 

85.02 

18.46 

84  .94 

18.83 

84.85 

19.20 

87 

88 

86.08 

18.30 

86.00 

18.67 

85.91 

19.05 

85.83 

19.42 

88 

89 

87.06 

18.50 

86.97 

18.88 

86.89 

15.26 

86.81 

19.64 

89 

90 

88.03 

18.71 

87.95 

19.10 

87.87 

19.48 

87.78 

19.86 

90 

91 

89.01 

18.92 

88.93 

19.31 

88.84 

19.70 

88.76 

20.08 

91 

92 

89.99 

19.13 

89.91 

19.52 

89.82 

19.91 

89.73 

20.30 

92 

93 

90.97 

19.34 

90.88 

19.73 

90.80 

20.13 

90.71 

20.52 

93 

94 

91 .9t 

19.54 

91.86 

19.94 

91.77 

20.35 

91.68 

20.75 

94 

95 

92.92 

19.75 

92.84 

20.16 

92.75 

20.56 

92.66 

20.97 

95 

96  i 

93.90 

19.96 

93.81 

20.37 

93.72 

20.78 

93.63 

21.19 

96 

97  | 

94.88 

20.17 

94.79 

20 . 58 

94.70  j 

20.99 

94  .61 

21.41 

97 

98  I 

95.86 

20.38 

95.77 

20.79 

95.68 

21.21 

95.58 

21.63 

98 

99  I 

96.84 

20.58 

96.75 

21.01 

96.65 

21.43 

96.56 

21.85  ! 

99 

100 

97.81 

20.79 

97.72 

21.22 

97.63  | 

21.64 

97.53 

22.07 

100 

6 

o 

a 

Dep.  | 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

© 

6 

a 

ui 

m 

Q 

78  Deg. 

7V  Deg 

77*  Deg. 

77*  Deg. 

eJ 

to 

5] 

08 


TRAVERSE  TABLE. 


o 

13  Deg. 

13*  Deg. 

13|  Deg. 

I3|  Deg. 

cl 

u 

8 

3 

o 

® 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

n 

a 

1 

0.97 

0.23 

0.97 

0.23 

“079T 

“0.23 

0.97 

0.24 

“T 

* 

1.95 

0.45 

1.95 

0.46 

1.95 

0.47 

1.94 

0.48 

2 

J 

2 92 

0.67 

2.92 

0.69 

2.92 

0.70 

2.91 

0.71 

3 

1 

3.90 

0.90 

3.89 

0.92 

3.89 

0.93 

3.89 

0.95 

4 

5 

4.87 

1.12 

4.87 

1.15 

4.86 

1.17 

•4.86 

1.19  1 

5 

6 

5.85 

1.35 

5.84 

1.38 

5.83 

1.40 

6.83 

1.43  ! 

6 

7 

6.82 

1 .57 

6.81 

1.60 

6.81 

1.63 

6.80 

1.66 

7 

ft 

7.80 

1.80 

7.79 

1.83 

7.78 

1.87 

7.77 

1.90 

9 

‘J 

8.77 

2.02 

8.76 

2.06 

8.76 

2.10 

8.74 

2.14 

9 

10 

9.74 

2.25 

9.73 

2.29 

9.72 

2.33 

9.71  1 

2.38 

10 

11 

10.72 

2.47 

10.71 

2.52 

10.70 

2.57 

10.68 

2.61 

11 

12 

11.69 

2.70 

11.68 

2.75 

11.67 

2.80 

11.66 

2.85 

12 

13 

12.67 

2.92 

12.65 

2.98 

12.64 

3.03 

12.63 

3.09 

13 

14 

13.64 

3.15 

13.63 

3.21 

13.61 

3.27 

13.60 

3.33 

14 

15 

14.62 

3.37 

14.60 

3.44 

14.59 

3.50 

14.57 

3.57 

15 

16 

15.59 

3.60 

15.57 

3.67 

15.56 

3.74 

15.54 

3.80 

16 

17 

16.57 

3.82 

16.55 

3.90 

16.53 

3.97 

16.51 

4.04 

17 

18 

17.54 

4.05 

17.52 

4.13 

17.50 

4.20 

17.48  1 

4.28 

18 

19 

18.51 

4.27 

18.49 

4.35 

18.48 

4.44 

18.46  1 

4.52 

19 

20 

19.49 

4.50 

19.47 

4.58 

19.45 

4.67 

19.43  1 

4.75 

20 

21  1 20.46 

4.72 

20.44 

4.81 

20.42 

4.90 

20.40 

4.99 

21 

22 ; 

21.44 

4.95 

21.41 

5.04 

21.39 

5.14 

21.37  1 

5.23 

22 

23 

22.41 

5.17 

22.39 

5.27 

22.36 

5.37 

22.34 

5.47 

23 

24 

23.38 

5.40 

23.36 

5.50 

23.34 

5.60 

23.31  j 

5.70 

24 

25 

24.36 

5.62 

24.33 

5.73 

24.31 

5.84 

24.28 

5.94 

25 

26 

25.33 

5.85 

25.31 

5.96 

25.28 

6.07 

25.25 

6.18 

26 

27 

26.31 

6.07 

26.28 

6.19 

26.25 

6.30 

26.23  ; 

6.42 

27 

28 

27.28 

6.30 

27.25 

6.42 

27.23 

j 6.54 

27.20 

6.66 

28 

29 

28.26 

6.52 

28.23 

6.65 

28.20 

i 6.77 

28.17 

6.89 

29 

30 

29.23 

6.75 

29.20 

6.88 

29.17 

1 7.00 

29.14 

7.13 

30 

31 

30.21 

6.97 

30.17 

7.11 

30.14" 

7.24 

30.11 

7.37 

31 

32 

31.18 

7.20 

31.15 

7.33 

31.12 

7.47 

31.08 

7.61 

32 

33 

32.15 

7.42 

32.12 

' 7.56 

32.09 

7.70 

32.05 

7.84 

33 

34 

33.13 

7.65 

33.09 

7.79 

33.06 

7.94 

33.03 

8.08 

34 

35 

34.10 

! 7.87 

34.07 

8.02 

34.03 

8.17 

34  00 

8.32 

35 

36 

35.08 

8.10 

35.04 

8.25 

35.01 

8.40 

34.97 

8.56 

36 

37 

36.05 

8.32 

36.02 

8.48 

35.98 

8.64 

35.94 

8.79 

37 

38 

37.03 

8.55 

36.99 

8.71 

36.95 

8.87 

36.91 

9.03 

j 38 

39 

38.00 

8.77 

37.96 

8.94 

37.92 

9.10 

37.88 

9.27 

39 

40 

38.97 

9.00 

38.94 

9.17 

38.89 

9.34 

38.85 

9.51 

i 40 

41 

39.95 

9.22 

39‘91 

9.40 

39.87 

9.57 

39.83 

9.75 

! 41 

42 

40.92 

9.46 

40.98 

9.63 

40.84 

9.80 

40.80 

9.98 

42 

43 

41.90 

9 67 

41.86 

9.86 

41.81 

10.04 

41.77 

10.22 

! 43 

44 

42.87 

9.90 

42.83 

10.08 

42.78 

10.27 

42.74 

! 10.46 

44 

45 

43.85 

.0.12 

43.80 

10.31 

48.76 

1 10.51 

43.71 

10.70 

45 

46 

44.82 

10.35 

44.78 

10.54 

44.73 

10.74 

44.68 

10.93 

46 

17 

45.80 

10.57 

45 . 75 

10.77 

45 . 70 

10.97 

45 . 65 

11.17 

47 

4ft 

46  77 

l 10.80 

46.72 

11.00 

46.67 

11.21 

46.62 

11.41 

48 

49 

47.74  i 11.02 

47.70 

11.23 

47.65 

11.44 

47.60 

11.66 

49 

50 

48.72 

11.25 

48.67 

11.46 

48.62 

1 11.67 

48^57 

H .88 

60 

i 

g 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| Lat. 

Dep. 

Lat. 

V 

u 

a 

3 

s 

Q 

1 77  Deg. 

76.f  Deg. 

Deg 

761  Deg. 

3 

m 

a 

o 

SB 

2 

a 

a 

tl 

52 

63 

54 

65 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

'81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

00 

© 

© 

e 

cJ 

« 

5 


rK AVERSE  TAELE. 


99 


13*  Deg. 

13-* 

Deg. 

13|  Deg. 

1 

O 

B' 

£ 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 Dep. 

a 

a 

® 

49.64 

11.69 

49.59 

11.91 

49.54 

12.12 

51 

50.62 

11.92 

50.56 

12. 14 

50.51 

J 2 .36 

' 52 

51.59 

12.15 

51.54 

12.37 

51.48 

12.60 

53 

52.56 

12.38 

52.51 

12.61 

52.45 

i 54 

53.54 

12.61 

53.48 

12.84 

53.42 

13.  07 

55 

54.51 

12.84 

54.45 

13.07 

54.40 

13.31  1 56 

55.48 

13.06 

55.43 

13.31 

55.37 

13.55 

1 57 

56.46 

13.29 

56.40 

13.54 

56  34 

13.79 

1 58 

57.43 

13.52 

57.37 

13.77 

57.31 

14  02 

59 

58.40 

13.75 

58.34 

14.01 

58.28 

14.  26 

60 

59.38 

13.98 

59.31 

"14.24 

59.25 

14.50 

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60.35 

14.21 

60.29 

14.47 

60.22 

14.74 

62 

61.32 

14.44 

61.26 

14.71 

61.19 

14.9/ 

63 

62.30 

14.67 

62.23 

14.94 

62. 17 

15.2 

64 

63.27 

14.90 

63.20 

15.17 

63.14 

15.45 

65 

64.24 

15.13 

64.18 

15.41 

64.11 

15.69 

66 

65.22 

15.36 

65.15 

15.64 

65.08 

15.93 

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66.19 

15.59 

66.12 

15.87 

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16.16 

68 

67.16 

15.81 

67.09 

16.11 

67.02 

16.40 

69 

68.14 

16.04 

68.07 

16.34 

67.99 

16.64 

70 

69.11 

16.27 

69.04 

16.57 

68.97 

16:88 

71 

70.08 

16.50 

70.01 

16.81 

69.94 

17.11 

72 

71.06 

16.73 

70.98 

17.04 

70.91 

17.35  , 

l 7Z 

72.03 

16.96 

71.96 

17.28 

71.88 

17.59 

1 74 

73.00 

17.19 

72.93 

17.50 

72.85 

17.82  1 

1 75 

73.98 

17.42 

73.90 

17.74 

73.82 

18.06 

76 

74.95 

17.65 

74.87 

17.98 

74.79 

18.30 

77 

75.92 

17.88 

75.84 

18.21 

75.76 

18.54 

78 

76.90 

18.11 

76.82 

18.4-1 

76.74 

18.78 

79 

77.87 

18.34 

77.79 

18.68 

77.71 

19.01 

8( 

78.84 

18.57 

78.76 

18.91 

78.68 

19.25 

81 

79.82 

18.79 

79.73 

19.14 

79.65 

19.49 

82 

80.79 

19.02 

80.71 

19.38 

80.62 

19.73 

83 

81 .76 

19.25 

81.68 

19.61 

81.59 

19.97 

84 

82.74 

19.48 

82.65 

19-84 

82.56 

20.20 

85 

83.71 

19.71 

83.62 

20-08 

83.54 

20.44 

86 

84.68 

19.94  ! 

84.60 

20.31 

84.51 

20.68 

87 

85.66 

20.17 

85.57  | 

20.54 

85.48 

20.92 

88 

86.63 

20.40 

86.54 

20.78 

86.45 

21.15 

89 

87.60 

20.63 

87.51 

21.01 

87.42 

21.39 

90 

88. 58 

20.86 

88.49 

21.24 

88.39 

21.63  | 

91 

89.55 

21.09 

89.46 

21.48 

89.36 

21 .87  1 

92 

90.52 

2’  32 

90.43 

21.71 

90.33 

22.10  ! 

1 93 

91.50 

21.54 

| 91.40 

21.94 

91.31 

22.34 

94 

92.47 

21.77 

92.38 

22.  18 

92.28 

22.58 

95 

93.44 

22.00 

93.35 

22.41 

93.25 

22.82 

96 

94.42 

22 . 23 

94.32  ' 

22.64 

94.22  ! 

23.06 

97 

95.39 

22.46 

95.29  ! 

22.88 

95.19 

23.29  , 

98 

96.36 

22.69 

96.26 

23.11 

96.16 

23.53 

99 

97.34 

22.92 

97.24 

23.34 

97.13 

23  77 

100 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

c 

76|  Deg. 

76* 

Dog. 

763  Deg. 

bJ 

en 

o 

100 


TRAVERSE  TA1*LI£. 


Distance.  1 

14  Dog. 

Lat.  1 Dep. 

14i  Deg. 

141 

Deg 

14|  Deg. 

Distance.1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 

97 

o. 

24 

0.97 

0.25 

0.97 

0.25* 

0. 

97 

0725 

1 

2 

1 

94 

0. 

48 

1.94 

0.49 

1.94 

0.50 

1. 

93 

0.51 

0 

3 

2. 

91 

0. 

73 

2.91 

0.74 

2.90 

0.75 

2. 

90! 

0.76 

3 

4 

3. 

88 

0. 

97 

3.88  i 

0.98 

3.87 

1.00 

3. 

87  i 

1.02 

4 

5 

4 

85 

1 

21 

4.85 

1 .23 

4.84 

1.25 

4. 

84  ! 

1.27 

5 

6 

5 

82 

1. 

45 

5.82 

1.48 

5.81 

1.50 

5. 

80  j 

1.53 

6 

7 

6 

79 

1 

69 

6.78 

1 72 

6.78 

1.75 

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77 

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7 

8 

7 

76 

1 

94 

7.75 

1.97 

7.75 

2.00 

7. 

74 

2.04  1 

8 

9 

8 

73 

2. 

18 

8.72 

2.22 

8.71 

2.25 

8. 

70 

2.29 

9 

10 

9 

70 

2. 

42 

9.69 

2.46 

9.68 

2.50 

9. 

67 

2.55 

10 

11 

10 

67 

2. 

66 

10.66 

2.71 

10.65 

2.75 

10. 

64 

’ 2.80 

11 

12 

11 

64 

O 

90 

11.63 

2.95 

11.62 

3.00 

11. 

60 

3.06 

12 

13 

12 

61 

3. 

15 

12.60 

3.20 

12.59 

3.25 

12. 

57 

3.31 

13 

14 

13 

58 

3 

39 

13.57 

3.45 

13.55 

3.51 

13. 

54 

3.56 

14 

15 

14 

55 

3 

63 

14.54 

3.69 

14.52 

3.76 

14. 

51 

3.82 

15 

16 

15 

52 

3 

87 

15.51 

3.94  1 

15.49 

4.01 

15. 

,47 

4.07 

16 

17 

16 

50 

4 

1 1 

16.43 

4.18  ! 

16.46 

4.26 

16. 

,44 

4.33 

17 

18 

17 

47 

4 

35 

17.45 

4.43  | 

17.43 

4.51 

17. 

41 

4.58 

18 

19 

18 

44 

4 

60 

18.42 

4.68  1 

18.39 

4.76 

18. 

.37 

4.84 

19 

20 

19 

41 

4 

84 

19.38 

4.92 

19.36  ! 

5.01 

19, 

.34 

5.09 

20 

21 

20 

38 

5 

08 

20.35 

5.17 

20.33 

i 5.26 

20, 

.31 

5.35 

21 

22 

i 2i 

.35 

5. 

,32 

21 .32 

5.42 

21.30 

i 5.51 

21, 

.28 

5.60 

22 

23 

i 22, 

.32 

5, 

.56 

22.29 

5.66 

22.27 

5.76 

22, 

.24 

5.86 

23 

24 

! 23, 

.29 

5, 

.81 

23 

5.91 

23.24 

| 6.01 

23, 

.21 

6.11 

24 

25 

! 24 

.26 

6. 

.05 

24.23 

6.15 

24.20 

6.26 

24, 

.18 

6.37 

25 

26 

25 

.23 

6, 

.29 

25.20 

6.40 

25.17 

6.51 

25 

.14 

6.62 

26 

27 

26 

.20 

6 

.53 

26.17 

6.65 

1 26.14 

6.76 

26 

.11 

6.87 

27 

28 

27 

.17 

6, 

.77 

27.14 

6.89  1 

27.11 

7.01 

27 

.08 

7.13 

28 

29 

28 

.14 

7, 

.02 

28.11 

7.14 

28.08 

7.26 

28 

.04 

7.38 

29 

30 

29 

.11 

i 7 

.26 

29.08 

7.38 

29.04 

7.51 

29 

.01 

7.64 

30 

31 

30 

.08 

! 7, 

.50 

30.05 

' 7.63 

30.01 

7.76 

29 

.98 

7.89 

31 

32 

31 

.05 

! 7 

.74 

31.02 

7.88 

30,98 

8.01 

30 

.95 

8.15 

32 

33 

32 

.02 

1 7 

.98 

31.98 

8.12 

31.95 

8.26 

31 

.91 

8.40 

33 

34 

32 

.99 

8 

.23 

32.95 

8.37 

32.92 

8.51 

32 

.88 

8.66 

34 

35 

33 

.96 

8 

.47 

33.92 

8.62 

33.89 

8.76 

33 

.85 

8.91 

35 

36 

34 

.93 

8 

.71 

34.89 

8.86 

34.85 

9.01 

34 

.81 

9.17 

36 

37 

36 

.90 

8 

.95 

35.86 

9.11 

35.82 

9.26 

35 

.78 

9.42 

37 

38 

36 

.87 

1 9 

. 19 

36.83 

9.35 

36.79 

9.51 

36 

.75 

9.67 

38 

39 

37 

.84 

9 

.44 

37.80 

9.60 

37.76 

9.7a 

37 

.71 

9.93 

39 

40 

38 

.81 

1 9 

.68 

38.77 

9.85 

38.73 

10.02 

38 

.68 

10.18 

40 

41 

39 

.78 

9 

.92 

39.74 

10.09 

39.69 

10.27 

39 

.65 

10.44 

41 

42 

40 

.75 

1 10 

.16 

40.71 

10.34 

40.66 

10.52 

40 

.62 

10.69 

42 

43 

4! 

.72 

10 

.40 

41 .68 

10.58 

41.63 

10.77 

41 

.58 

10.95 

43 

44 

,42 

.69 

10 

.64 

42.65 

10.83 

42.60 

11.02 

42 

.55 

11.20 

44 

45 

43 

.66 

10 

.89 

43.62 

11.08 

43.57 

11.27 

43 

.52 

11.46 

45 

46 

44 

.63 

11 

.13 

44.58 

11.32 

44.53 

11.52 

44 

.48 

11.71 

46 

47 

45 

.60 

11 

.37 

45.55 

11.57 

45.50 

11.77 

45 

.45 

11  97 

47 

48 

46 

.57 

11 

.61 

[ 46.52 

11.82 

46.47 

12.02 

46 

.42 

12.22 

i 48 

49 

47 

.54 

11 

.85 

47.49 

12.06 

47.44 

12.27 

47 

.39 

12  48 

49 

50 

48 

.61 

ija 

.10 

48.46 

12.31 

48.41 

12.52 

48 

.35 

12  73 

50_ 

8 

5 

■ 

5 

V 

Dep 

, Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D 

ep. 

Lat. 

d 

6 

a 

3 

2 

a 

| 

76  Deg. 

75 } Deg. 

76  \ Deg. 

75}  Deg. 

Til  AVERSE  TABLE. 


101 


Distanco. 

14  Dog. 

14i  Deg. 

14$  Deg. 

14|  Deg. 

C 

5! 

§ 

o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| Dep. 

51 

49.49 

12.34 

49.43 

12.65 

49738 

12.77 

49732 

1 12798 

|*51 

52 

50.46 

12.58 

50.40 

12.80 

50.34 

13.02 

50.29 

. 13.24 

I 52 

53 

51.43 

12.82 

51.37 

13.05 

51.31 

13.27 

51.25 

13.49 

53 

54 

52.40 

13.06 

52.34 

13.29 

52.28 

13.52 

52.22 

13.75 

1 54 

55 

53.37 

13.31 

53.31 

13.54 

53.25 

13.77 

53.19 

14.00 

| 55 

56 

54.34 

13.55 

64.28 

13.78 

54.22 

14.02 

54.15 

14.26 

! 66 

5r’ 

55.31 

13.79 

55.25 

14.03 

55.18 

14.27 

55.12 

14.51 

1 57 

58 

56.28 

14.03 

56.22 

14.28 

56.15 

14.52 

56.09 

14.77 

! 58 

59 

57.25 

14.27 

57.18 

14.52 

57.12 

14.77 

57.06 

15.02 

59 

60 

58.22 

14.52 

58.15 

14.77 

58.09 

15.02 

58.02 

15.28 

60 

61 

59.19 

14.76 

59.12 

15.02 

59.06 

15.27 

58.99 

15.53 

! 61 

62 

60.16 

15.00 

60.09 

15.26 

60.03 

15.52 

59.96 

15.79 

62 

63 

61.13 

15.24 

61.06 

15.51 

60.99 

15.77 

60.92 

16.04 

i 63 

64 

62.10 

15.48 

62.03 

15.75 

61.96 

16.02 

61.89 

16.29 

64 

65 

63.07 

15.72 

63.00 

16.00 

62 . 93 

16.27 

62.86 

16.55 

1 65 

66 

64.04 

15.97 

63.97 

16.25 

63.90 

16.53 

63.83 

16.80 

66 

67 

65.01 

16.21 

64.94 

16.49 

64.87 

16.78 

64.79 

17.06 

67 

68 

65.98 

16.45 

65.91 

16.74 

65.83 

17.03 

65.76 

17.31 

68 

69 

66.95 

16.69 

66.88 

16.98 

66.80 

17.28 

66.73 

17.57 

69 

70 

67.92 

16.93 

67.85 

17.23 

67.77 

17.53 

67.69 

17.82 

70 

71 

68.89 

17.18 

68.82 

17.48 

68.74 

17.78 

68.66 

18.08 

71 

72 

69.86 

17.42 

69.78 

17.72 

69.71 

18.03 

69.63 

18.33 

72 

73 

70.83 

17.66 

70.75 

17.97 

70.67 

18.28 

70.59 

18.59 

73 

74 

71.80 

17.90 

71.72 

18.22 

71.64 

18.53 

71.56 

18.84 

74 

75 

72  77 

18.14 

72.69 

18.46 

72.61 

18.78 

72.53 

19.10 

75 

76 

73 . 74 

18.39 

73.66 

18.71 

73.58 

19.03 

73.50 

19.35 

76 

77 

74.71 

18.63 

74.63 

18.95 

74.55 

19.28 

74.46 

19.60 

77 

78 

75.68 

18.87 

75.60 

19.20 

75.52 

19.53 

75.43 

19.86 

78 

79 

76.65 

19.11 

76.57 

19.45 

76.48 

19.78 

76.40 

20.11 

79 

80 

77.62 

19.35 

77.54 

19.69 

77.45 

20.03 

77.36 

20.37 

80 

81 

78.59 

19.60 

78.51 

19.94 

78.42 

20.28 

78.33 

20.62 

Ti 

82 

79.56 

19.84 

79.48 

20.18 

79.39 

20.53 

79.30 

20.88 

82 

83 

80.53 

20.08 

80.45 

20.43 

80.36 

20.78 

80.26 

21.13 

83 

84 

81.50 

20.32 

81.42 

20.68 

81.32 

21.03 

81 .23 

21.39 

84 

85 

82.48 

20.56 

82.38 

20.92 

82.29 

21 .28 

82.20 

21.64 

85 

86 

83.45 

20.81 

83.35 

21.17 

83.26 

21.53 

83.17 

21 .90 

86 

87 

84.42 

21.05 

84.32 

21.42 

84.23 

21  .78 

84.13 

22. 15 

87 

88 

85.39 

21.29 

85  29 

21.66 

85.20 

22.03 

85.10 

22.41 

88 

89 

I 86.36 

21.53 

86.26 

21.91 

86.17 

22.28 

86.07 

22.66 

89 

90 

! 87.33 

21.77 

87.23 

22.15 

87.13 

22.53 

87.03 

22.91 

90 

91 

1 88.30 

22.01 

88.20 

22.40 

88.10 

22.78 

88.00 

23.17 

91 

92 

' 89.27 

22.26 

89.17 

22.65 

89.07 

23.04 

88.97 

23.42 

92 

93 

90.24 

22.50 

90.14 

22.89 

90.04 

23.29 

89.94 

23.68 

93 

94 

91  21 

22.74 

91.11 

23.14 

91.01 

23.54 

90.90 

23.93 

94 

95 

92  18 

i 22.98 

92.08 

23.38 

91.97 

23.79 

91.87 

24.19  i 

95 

9t 

93  15 

23.22 

93.05 

23.63 

92.94 

24.04 

92.84 

24.44 

96 

97 

94.  12 

23.47 

94.02 

23.88 

93.91 

24.29 

93.80 

24.70 

97 

08 

95.09 

! 23.71 

94.98 

24.12 

94.88 

24.54 

94.77 

24.95 

98 

99 

96.06 

23  95 

95.95 

24.37 

95.85 

24.79 

95.74 

25.21 

99 

IOC 

07.03 

24-19 

96.92 

24.62 

96.81 

25.04 

96.70 

25.46 

100 

| 

Dep. 

1 Lai. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

s' 

c 

P 

76  Deg. 

75f  Deg 

75$  Deg. 

75$  Deg 

CCS 

5 

i 

102 


TRAVERSE  TABLE. 


o 

5* 

15  Deg. 

154  Deg. 

Deg. 

16|  Deg 

>P 

Distance. 

3 

« 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dt 

■ 1 

0.97 

0.26 

0.90 

~~0.26 

0.96 

0.27 

0 . 96 

0. 

27 

1 

2 

1.93 

0.52 

1.93 

0.53 

1.93 

0.53 

1.92 

0. 

54 

2 

3 

2.90 

0.78 

2.89 

0.79 

2.89 

0.80 

2.89 

0. 

81 

3 

l 

3.86 

1.04 

3.86 

1.05 

3.85 

1.07 

3.85 

1, 

09 

4 

5 

4.83 

1.29 

4.82 

1.32 

4.82 

1.34 

4.81 

1. 

36 

5 

i 

6.80 

1.55 

5.79 

1.58 

6.78 

1.60 

5.77 

1. 

63 

6 

7 

6.76 

1.81 

6.75 

1.84 

6.75 

1.87 

6.74 

1. 

90 

7 

8 

? 73 

2.07 

7.72 

2.10 

7.71 

2.14 

7.70 

2. 

17 

8 

9 

8.69 

2.33 

8.68 

2.37 

8.67 

2.41 

8.66 

2. 

44 

9 

10 

9.66 

2.59 

9.65 

2.63 

9.64 

2.67 

9.62 

2. 

71 

10 

11 

10.63 

2.85 

10.61 

2.89 

10.60 

2.94 

10.59 

2. 

99 

11 

12 

11.59 

3.11 

11.58 

3.16 

11.56 

3.21 

11.55 

3. 

26 

12 

13 

12.56 

3.36 

12.54 

3.42 

12.53 

3.47 

12.51 

3. 

53 

13 

14 

13.52 

3.62 

13.51 

3.68 

13.49 

3.74 

13.47 

3. 

80 

14 

15 

14.49 

3.88 

14.47 

3.95 

14.45 

4.01 

14.44 

4. 

07 

15 

16 

15.45 

4.14 

15.44 

4.21 

15.42 

4.28 

J5.40 

4. 

34 

16 

17 

16.42 

4.40 

16.40 

4.47 

16.38 

4.54 

16.36 

4. 

61 

17 

18 

17.39 

4.66 

17.37 

4.73 

17.35 

4.81 

17.32 

4. 

89 

18 

19 

18.35 

4.92 

'8.33 

5.00 

18.31 

5.08 

18.29 

5. 

16 

19 

20 

19.32 

5.18 

19. 3C 

5.26 

19.27 

5.34 

19.25 

5. 

43 

20 

21 

20.28 

5.44 

20.26 

5.52 

‘20.24 

5.61 

20.21 

5. 

"70 

21 

22 

21.25 

5 . 69 

21.23 

5.79 

21.20 

5.88 

21.17 

5. 

,97 

22 

23 

22.22 

5.95 

22. 19 

6.05 

22.16 

6.15  j 

22.14 

6. 

,24 

23 

24 

23. 18 

, 6.21 

23.15 

| 6.31 

23.13 

6.41 

23.10 

6. 

,51 

24 

25 

24.15 

6.47 

24. 12 

6.58 

24.09 

6.68 

24.06 

6. 

,79 

25 

28 

25.11 

6.73 

25.08 

6.84 

25.05 

6.95  | 

25.02 

7. 

,06 

26 

27 

26.08 

6.99 

26.05 

7.10 

26.02 

7.22  ! 

25.99 

7, 

,33 

27 

28 

27.05 

7.25 

27.01 

7.36 

26.98 

7.48  ; 

26.95 

7, 

,60 

28 

29 

28.01 

7.51 

27.98 

7.63 

27.95 

7.75 

27.91 

7, 

,87 

29 

30 

28.98 

7.76 

28-.  94 

7.89 

28.91 

8.02 

28.87 

8. 

,14 

30 

31 

29.94 

8.02 

29.91 

8.15 

29.87 

8.28 

29.84 

8. 

,41 

31 

32 

30.91 

8.28 

30.87 

8.42 

30.84 

8.55 

30.80 

8. 

,69 

32 

33 

31.88 

8.54 

31.84 

8.68 

31.80 

8.82 

31.76 

8, 

,96 

33 

34 

32.84 

8.80 

32.80 

8.94 

32.76 

9.09 

32.72 

9. 

,23 

34 

35 

33.81 

9.06 

33.77 

9.21 

33.73 

9.35 

33.69 

9. 

,59 

35 

36 

34.77 

9.32 

34.73 

9.47 

34.69 

9.62 

34.65 

9. 

.77 

36 

37 

35.74 

9.58 

35.70 

9.73 

35.65 

9.89 

35.61 

10, 

.04 

37 

38 

36.71 

9.84 

36.66 

10.00 

36.62 

10.16 

36.57 

10 

.31 

38 

39 

37.67 

10.09 

37.63 

10.26 

37.58 

10.42 

37.54 

10, 

.59 

39 

40 

38.64 

10.35 

38.59 

10.52 

38.55 

10.69 

38.50 

i 10 

.86 

10 

4 1 

39.60 

10.61 

"39.56 

10.78 

39.51 

10.96 

39.46 

1 1 

.13 

11 

42 

40.57 

10.87 

40.52 

11.05 

40.47 

11.22 

40.42 

1 11 

.40 

42 

43 

i 41.53 

11.13 

41.49 

11.31 

41.44 

11.49 

41.39 

! 11 

.67 

43 

44 

I 42.50 

11.39 

42.45 

11.67 

42.40 

11.76 

42.35 

11 

.94 

14 

45 

43.47 

! 11.65 

43.42 

11.84 

43.36 

12.03 

43.31 

12 

.21 

45 

16 

14.43 

11.91 

44.38 

, 12.10 

44.33 

l 12.29 

144.27 

i 12 

.49 

46 

47 

45.40 

12.16 

45  35 

12.36 

45.29 

12.56 

' 45.24 

12 

76 

47 

48 

46.36 

12.42 

46.31 

12.63 

46.25 

12.83 

1 46.20 

13 

.03 

48 

49 

47.33 

12.68 

47.27 

12.89 

47.22 

13.09 

147.16 

13 

.30 

49 

50 

, 48.30 

12.94 

48.24 

13.15 

48.18 

13.36 

48 . 12_ 

13 

.57 

50 

Distance,  j 

Dep. 

Lat. 

Dep. 

* 

1 Lat 

Dep. 

Lat. 

1 Dep. 

| Lat. 

ID 

0 

1 

Q 

75  Dog. 

74}  Deg. 

74,i  Deg. 

74<j  Deg. 

o 

«' 

p 

3 

o 

® 

61 

62 

63 

64 

56 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

© 

S 

C 

a 

.9 

Q 


TRAVER8E  TABLE 


103 


15  Deg. 

15} 

Deg 

154 

Deg. 

15} 

Deg 

g 

| 

L 

at. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

p 

o 

(C 

49 

.26 

13 

20 

49 

20 

13 

.41 

49. 

.15 

\lL63 

49V0<r 

13. 

84 

50 

.23 

13 

46 

50 

17 

13 

.68 

50, 

.11 

13.90 

50.05 

14. 

11 

52 

51 

.19 

13 

72 

51 

13 

13, 

.94 

51, 

.07 

14. 16 

51.01 

14. 

39 

53 

52 

16 

13 

98 

52 

10 

14, 

.20 

52. 

.04 

14.43 

51.97 

14. 

60 

64 

53 

13 

14 

24 

53 

06 

14. 

.47 

53, 

,00 

14.70 

52.94 

14. 

93 

5 1 

54 

09 

14 

49 

64 

03 

14, 

.73 

53. 

.96 

14.97 

63.90 

15. 

20 

56 

55 

06 

14 

75 

54 

99 

14, 

.99 

54. 

,93 

15.23 

54 . 86 

15. 

47 

57 

56 

02 

15 

01 

55 

96 

15, 

.26 

55. 

.89 

15.50 

55.82 

15. 

74 

58 

56 

99 

15 

27 

56 

92 

15. 

,52 

56. 

,85 

15.77 

56.78 

16. 

01 

59 

57 

96 

15 

53 

57 

89 

15. 

,78 

57. 

.82 

16.03 

57.75 

16. 

29 

60 

58 

92 

15 

79 

58 

85 

16, 

.04 

58. 

.78 

16.30 

58.71 

16. 

56 

61 

59 

89 

16 

05 

59 

82 

16, 

,31 

59, 

.75 

16.57 

59.67 

16. 

83 

62 

60 

85 

16 

31 

60 

78 

16, 

,57 

60, 

,71 

16.84 

60.63 

17. 

10 

63 

61 

82 

16 

56 

61 

75 

16, 

,83 

61 , 

.67 

17. 10 

61.60 

17. 

37 

64 

62 

79 

16 

82 

62 

71 

17. 

,10 

62. 

.64 

17.37 

62.56 

17. 

64 

65 

63 

75 

17. 

08 

63 

68 

17, 

.36 

63, 

,60 

17.64 

63.52 

17. 

92 

66 

64 

72 

17 

34 

64 

64 

17. 

.62 

64. 

,56 

17.90 

64.48 

18. 

19 

67 

65 

68 

17. 

60 

65 

61 

17. 

.89 

65, 

.53 

18.17 

65.45 

18. 

40 

68 

66 

65 

17. 

86 

66 

57 

18. 

.15 

66. 

,49 

18.44 

66.41 

18. 

73 

69 

67 

61 

18. 

12 

67 

54 

18, 

,41 

67. 

.45 

18.71 

67.37 

19. 

00 

70 

68 

58 

18. 

38 

68 

50 

18, 

.68 

68 , 

.42 

18.97 

68.33 

19. 

27 

71 

69 

55 

18. 

63 

69 

46 

18. 

,94 

69. 

,38 

19.24 

69 . 30 

19. 

54 

72 

70, 

.51 

18. 

89 

70. 

.43 

19. 

,20 

70, 

,35 

19.51 

70.26 

19. 

82 

73 

71, 

.48 

19. 

15 

71. 

,39 

19. 

,46 

71. 

31 

19.78 

71.22 

20. 

09 

74 

72, 

.44 

19. 

41 

72, 

.36 

19. 

.73 

72, 

.27 

20.04 

72.18 

20. 

36 

75 

73, 

.41 

19. 

67 

73, 

.32 

J 9 . 

.99 

73, 

.24 

20.31 

73. 15 

20. 

63 

76 

74 

.38 

19. 

93 

74, 

,29 

20, 

.25 

74, 

,20 

20.58 

74.11 

20. 

90 

77 

75, 

.34 

20. 

19 

75. 

.25 

20, 

,52 

75, 

.16 

20.84 

75.07 

21. 

17 

78 

70, 

.31 

20. 

45 

76, 

.22 

20. 

.78 

76, 

.13 

21.11 

l 76.03 

21 . 

44 

79 

77, 

.27 

20. 

71 

77, 

.18 

21, 

.04 

77, 

.09 

21.38 

77.00 

21. 

72  ! 

80 

78, 

.24 

20. 

,96 

78. 

,15 

21, 

,31  1 

78, 

.05 

21.65 

77.96 

21. 

99 

s* 

79, 

,21 

21. 

22 

79, 

,11 

21, 

.57 

79, 

.02 

21.91 

78.92 

22. 

26 

82 

80, 

.17 

21. 

48 

80, 

.08 

21. 

,83 

79, 

.98 

22.18 

79.88 

22. 

53 

83 

81, 

.14 

21 . 

,74 

81, 

.04 

22, 

.09 

80, 

.94 

22.45 

80.85 

22. 

80 

84 

82, 

.10 

22. 

00 

82, 

.01 

22, 

.36 

81  , 

.91 

22.72 

81.81 

23. 

07 

85 

83, 

.07 

22. 

,26 

82, 

.97 

22, 

.62 

82, 

.87 

22.98 

82.77 

23, 

34 

86 

84, 

.04 

22. 

.52 

83, 

.94 

22, 

.88 

83 

.84 

23.25 

83.73 

23 

62  1 

87 

85, 

.00 

22. 

,78 

84, 

.90 

23 

.15 

84 

.80 

23 . 52 

84.70 

23 

89 

88 

85 

.97 

23. 

,03 

85, 

.87 

23 

.41 

85 

.76 

23.78 

85 . 66 

24 

16 

89 

86 

.93 

23, 

.29 

86 

.83 

23 

.67 

86 

.73 

24.05 

86.62 

24 

43 

90 

87 

.90 

23, 

.55 

87 

.80 

23 

.94 

87 

769 

24.32 

87.58 

24. 

70 

91 

88 

.87 

23 

.81 

88 

.76 

24 

.20 

88 

.65 

24.59 

88.55 

24. 

,97 

92 

89 

.83 

24, 

.07 

89 

.73 

24 

.46 

89 

.62 

24.85 

89.51 

25. 

,24  1 

I 93 

90 

.80 

24 

.33 

90 

.69 

24 

.72 

90 

.58 

25.12 

90.47 

25. 

,52  | 

1 94 

91 

. 76 

24 

.59 

91 

.65 

24 

.99 

91 

34 

25 . 39 

91.43 

25. 

.79 

95 

92 

73 

24 

.85 

92 

.62 

25 

.25 

92 

.51 

; 25.65 

92.40 

26. 

,06 

1 90 

93 

.69 

25 

.11 

93 

.58 

25 

.51 

93 

.47 

| 25.92 

93.36 

26, 

.33 

97 

94 

.66 

25 

.36 

94 

.55 

25 

.78 

94 

.44 

| 26.19 

1 94.32 

26. 

.60 

98 

95 

.63 

25 

.62 

95 

.51 

26 

.04 

95 

.40 

| 26.46 

i 95.28 

26, 

.87 

99 

96 

.59 

25 

.88 

96 

.48 

26 

.30 

96 

.36 

26.72 

96 . 25 

27 

.14 

100 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

V 

o 
, C 

j 75  Deg. 

74} 

II 

Deg. 

744 

Deg. 

74} 

Deg. 

! i 

104 


traveksk  table. 


g 

ta 

< r ♦ 

16  Deg. 

16}  Deg. 

16£  Deg. 

1 

16?  Deg  | 

g 

5 

s 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep.  * 

5 

? 

1 

0.06 

0.28 

0.96 

0.28 

0.96 

~0  28 

0.96 

0.29 

l 

2 

1 92 

0 55 

1.92 

0.56 

1.92 

0.57 

1.92 

0.58 

2 

3 

2.88 

C 83 

2.88 

0.84 

2.88 

0.85 

2.87 

0.86 

3 

4 

3.85 

1.10 

3.84 

1.12 

3.84 

1.14 

3.83 

1.15 

4 

5 

4.81 

1 .38 

4.80 

1.40 

4.79 

1.42 

4.79 

1.44 

5 

6 

5.77 

1.65 

5.76 

1.68 

5.75 

1.70 

5.75 

1.73 

6 

7 

6.73 

1.93 

6.72 

1.96 

6.71 

1.99 

6.70 

2.02 

7 

8 

7.69 

2.21 

7.68 

2.24 

7.67 

2.27 

7.66 

2.31 

8 

9 

8.65 

2.48 

8.64 

2.52 

8.63 

2.56 

8.62 

2.59 

9 

10 

9.01 

2.76 

9.60 

2.80 

9.59 

2.84 

9.58 

2.88 

10 

11 

10.57 

3.03 

10.56 

3.08 

10.55 

3.12 

10.53 

3.17  1 

11 

12 

11.54 

3.31 

11.52 

3.36 

11.51 

3.41 

11.49 

3.46 

12 

13 

12.50 

3.58 

12.48 

3.64 

12.46 

3.69 

12.45 

3.75 

13 

14 

13.46 

3.86 

13.44 

3.92 

13.42 

3.98 

13.41 

4.03 

14 

15 

14.42 

4.13 

14.40 

4.20 

14.38 

4.26 

14.36 

4.32 

15 

16 

15.38 

4.41 

15.36 

4.48 

15.34 

4.54 

15.32 

4.61 

16 

17 

16.34 

4.69 

16.32 

4.76 

16.30 

4.83 

16.28 

4.90 

17 

18 

17.30 

4.96 

17.28 

5.04 

17.26 

5.11 

17.24 

5.19 

18 

19 

18.26 

5.24 

18.24 

5.32 

18.22 

5.40 

18.19 

5.48 

19 

20 

19.23 

5.51 

19.20 

5.60 

19.18 

5.68 

19. 15 

5.76 

20 

21 

20.19 

5.79 

20.16 

5.88 

20.14 

5.96 

20.11 

6.05 

21 

22 

21.15 

6.06 

21.12 

6.16 

21.09 

6.25 

21.07 

6.34 

22 

23 

22.11 

6.34 

22.08 

6.44 

22.05 

6.53 

22.02 

6.63 

23 

24 

23.07 

6.62 

23.04 

6.72 

23.01 

6.82 

22.98 

6.92 

24 

25 

24.03 

6.89 

24.00 

7.00 

23.97 

7.10 

23.94 

7.20 

25 

26 

24.99 

7.17 

24.96 

7.28 

24.93 

7.38 

24.90 

7.49 

26 

27 

25 . 95 

7.44 

25.92 

7.56 

25.89 

7.67 

25.85 

7.78 

27 

28 

26.92 

7.72 

26.88 

7.84 

26.85 

7.95 

26.81 

8.07 

28 

29 

27.88 

7.99 

27.84 

8.11 

27.81 

8.24 

27.77 

8.36 

29 

30 

28.84 

8.27 

28.80 

8.39 

28.76 

8.52 

28.73 

8.65 

30 

31 

29.80 

8.54 

29.76 

8.67 

29.72 

8.80 

29.68 

8.93 

31 

32 

30 . 76 

8.82 

30.72 

8.95 

30.68 

9.09 

30.64 

9.22 

32 

33 

31.72 

9.10 

31.68 

9.23 

31.64 

9.37 

31.60 

9.51 

33 

34 

32.88 

9.37 

32.64 

9.51 

32.60 

9.66 

32.56 

9.80 

34 

35 

33.64 

9.65 

33.60 

9.79 

33.56 

9.94 

33.51 

10.09 

35 

36 

34.61 

9.92 

34.56 

10.07 

34.52 

10.22 

34.47 

10.38 

36 

37 

35.57 

10.20 

35  52 

10.35 

35.48 

10.51 

35.43 

10.66 

37 

38 

36.53 

10.47 

36.48 

10.63 

36.44 

10.79 

36.39 

10.95 

38 

39 

37.49 

10.75 

37-44 

10.91 

37.39 

11.08 

37.35 

11.24 

39 

40 

38.45 

11.03 

38.40 

11.19 

38.35 

11.36 

38.30 

11.53 

40 

41 

39.41 

11.30 

39.36 

11.47 

39.31 

11.64 

39.26 

11.82 

41 

42 

40.37 

11.58 

40.32 

11.75 

40.27 

11.93 

40.22 

12.10 

42 

43 

41.33 

11.85 

41.28 

12.03 

41  23 

12.21 

41.18 

12.39 

43 1 

41 

42.30 

12.13 

42.24 

12.31 

42  19 

12.50 

42.13 

12.68 

i 44 

15 

43.26 

12.40 

43.20 

12.59 

43.  15 

12.78 

43.09 

12.97 

45 

46 

14.22 

12.68 

44. 16 

12.87 

44.11 

13.06 

44.05 

13.26 

46 

47 

45.18 

12.95 

45. 12 

13.15 

45.06 

13.35 

45.01 

13.55 

47 

48 

46.  14 

13.23 

46.08 

13.43 

46.02 

13.63 

45 . 96 

13.83 

48 

49 

47. 10 

13.51 

47.04 

,,.71 

46.98 

13.92 

46.92 

14.12 

49 

50 

48.06 

I :i . 78 

48.00 

13.99 

47.94 

14.20 

47.88 

14.41 

50 

| 

Dep 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

o 

a 

9 

Q 

74 

i 

Dog. 

73?  Deg. 

7*4  Deg. 

73}  Dog. 

3 

.a 

Q 

TRAVERSE  TABLE. 


105 


u 

B‘ 

S' 

16  Deg. 

164  Deg. 

16A 

Deg 

16f  Deg. 

C 

5’ 

fO 

§ 

§ 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

£3 

o 

o 

61 

49.02 

14.06 

48.96 

14.27 

48.90 

14748 

48.84 

14770 

51 

52 

49.99 

14.33 

49.92 

14.55 

49  86  ! 

14.77 

49.79 

14.99 

52 

63 

50.95 

14.61 

50.88 

14  83 

50.82  1 15.05 

50.75 

15.27 

53 

54 

51.91 

14.88 

51.84 

15  11 

51.78 

15.34 

51.71 

15.56 

54 

55 

52.87 

15.16 

52.80 

15.39 

52.74 

15.62 

52.67 

15.85 

55 

56 

53.83 

15,44 

53.76 

15.67 

53.69 

15.90 

53.62 

16.14 

56 

57 

54.79 

15.71 

54.72 

15.95 

54.65 

16.19 

54.58 

16.43 

57 

68 

55.75 

15.99 

55.68 

16.23 

55.61 

16.47 

55.54 

16.72 

58 

59 

56.71 

16.26 

56.64 

16.51 

56.57 

16.76 

56.50 

17.00 

59 

60 

57.68 

16-54 

57.60 

16.79 

57.53 

17.04 

57.45 

17.29 

60 

61 

58.64 

16.81 

58.56 

17.07 

58.49 

17.32 

58.41 

17.58 

'61 

62 

59.60 

17.09 

59.52 

17.35 

59.45 

17.61 

59.37 

17.87 

62 

63 

60.56 

17.37 

50  48 

17.63 

60.41 

17.89 

60.33 

18.16 

63 

64 

61.52 

17.64 

61 .44 

17.91 

61.36 

18.18 

61.28 

18.44 

64 

65 

62.48 

17.92 

62.40 

18.19 

62.32 

18.46 

62.24 

18.73 

65 

66 

63.44 

18.19 

63.36 

18.47 

63.28 

18.74 

63.20 

19.02 

60 

67 

64.40 

18.47 

64.32 

18.75 

64.24 

19.03 

64.16 

19.31 

67 

68 

65.37 

18.74 

65.28 

19.03 

65.20 

19.31 

65.11 

19.60 

68 

69 

66.33 

19.02 

66.24 

19.31 

66.16 

19.60 

66.07 

19.89 

69 

70 

67.29 

19.29 

67.20 

19.59 

67.12 

19.88 

67.03 

20.17 

70 

71 

68.25 

19.57 

68. 16 

19.87 

68.08 

20.17 

67.99 

20.46 

71 

72 

69.21 

19.85 

69.12 

20.15 

69.03 

20.45 

68.95 

20.75 

72 

73 

70.17 

20.12 

70.08 

20.43 

69.99 

20.73 

69.90 

21.04 

73 

74 

71.13 

20.40 

71.04 

20.71 

70.95 

21  02 

70.86 

21.33 

74 

75 

72.09 

20.67 

72.00 

20.99 

71.91 

21.30 

71.82 

21 .61 

75 

76 

73.06 

20.95 

72.96 

21.27 

72.87 

21.59 

72.78 

21.90 

76 

77 

74.02 

21.22 

73.92 

21.55 

73.83 

21.87 

73.73 

22.19 

77 

78 

74.98 

21.50 

74.88 

21.83 

74.79 

22.15 

74.69 

22.48 

78 

79 

75.94 

21.78 

75.84 

22.11 

75.75 

22.44 

75 . 65 

22.77 

79 

8.0 

76.90 

22.05 

76.80 

22.39 

76.71 

22.72 

76.61 

23.06 

80 

81 

77.80 

22.33 

77.76 

22.67 

77.66 

23.01 

77.56 

23.34 

81 

82 

78.82 

22.60 

78.72 

22.95 

78.62 

23.29 

78.52 

23.63 

82 

83 

79.78 

22.88 

79.68 

23.23 

79.58 

23.57 

79.48 

23.92 

83 

84 

80.75 

23.15 

80.64 

23 . § 1 

80.54 

23.86 

80.44 

24.21 

84 

85 

81.71 

23.43 

81.60 

23.79 

81.50 

24.14 

81.39 

24.50 

85 

86 

82.67 

23.70 

82.56 

24.07 

82.46 

24.43 

82.35 

24.78 

86 

87 

83.63 

23.98 

83.52 

24.35 

83.42 

24.71 

83.31 

25.07 

87 

88 

84.59 

24.26 

84.48 

24.62 

84.38 

24.99 

84.27 

25.36 

88 

89 

85.55 

24.53 

85.44 

24.90 

85.33 

25.28 

85.22 

25.65 

89 

9C 

86.51 

24.81 

86.40 

25.18 

86.29 

25.56 

86.18 

25.94 

90 

91 

| 87.47 

25.08 

87.36 

25.46 

87.25 

25.85 

87.14 

26.23 

'91 

92 

1 88.44 

25.36 

88.32 

25.74 

88.21 

26.13 

88.10 

26.51 

92 

93 

i 89.40 

25.63 

89.28 

26.02 

89.17 

26.41 

89.05 

26.80 

9.3 

94 

90.36 

25.91 

90.24 

26.30 

90.13 

26.70 

90.01 

27.09 

94 

95 

91.32 

26.19 

91.20 

26.58 

91.09 

26.98 

90.97 

27.38 

95 

96 

92.28 

126.46 

92.16 

26.86 

92.05 

27.27 

91.93 

27.67 

96 

97 

93.24 

1 26.74 

93.12 

27.14 

93.01 

27.55 

92.88 

27.95 

97 

98 

94.20 

i 27.01 

94.08 

27.42 

93.96 

27.83 

93.84 

28.24 

98 

99 

95.16 

27.29 

95.04 

27.70 

94.92 

28.12 

94.80 

28.53 

99 

100 

96. 13^ 

27.56 

96.00 

27\98^ 

95.88 

28.40 

95.76 

28.82 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

8 

c 

3 

.a 

Q 

74  Deg. 

73|  Deg. 

73± 

Deg. 

Deg. 

s 

X 

1 S 

106 


TRAVERSE  TARLId. 


Distance.! 

17  Deg. 

17*  Deg. 

17*  Dog. 

17*  Deg. 

V 

5 

i 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

1 

0 

96 

0. 

29 

0. 

95 

0. 

30 

6795 

0 . 30 

0.95 

0.30 

2 

1 

91 

0. 

58 

1. 

01 

0. 

59 

1.91 

0.60 

1.90 

0.61 

a 

3 

2. 

87 

0. 

88 

2. 

87 

0. 

89 

2.86 

0.90 

2.86 

0.91 

3 

4 

3. 

83 

1. 

17 

3. 

82 

1. 

19 

3.81 

1.20 

3.81 

l 22 

4 

6 

1 

78 

1 

46 

4. 

78 

1 . 

48 

4.77 

1.50 

4.76 

1.52 

5 

0 

5. 

74 

1 

75 

5. 

73 

1. 

78 

5.72 

1.80 

5.71 

1.83 

6 

7 

6. 

69 

2. 

05 

6 

69 

2. 

08 

6.68 

2.10 

6.67 

2 13 

7 

8 

7 

65 

2. 

34 

7. 

64 

2. 

37 

7.63 

2.41 

7.62 

2.44 

9 

9 

8. 

61 

2. 

63 

8. 

60 

2. 

67 

8.58 

2.71 

8 57 

2.74 

9 

10 

9. 

56 

2. 

92 

9. 

55 

2. 

97 

9.54 

3.01 

9.52 

3.05 

10 

11 

10. 

52 

3. 

22 

10. 

51 

3. 

26 

10.49 

3.31 

10.48' 

3.35 

11 

12 

11. 

48 

3. 

51 

11. 

46 

3. 

56 

11.44 

3.61 

11.43 

3.66 

12 

13 

12 

43 

3. 

80 

12. 

42 

3. 

85 

12.40 

3.  91 

12.38 

3.96 

13 

14 

13 

39 

4. 

09 

13 

37 

4. 

15 

13.35 

4. 21 

13.33 

4.27 

14 

15 

14. 

34 

4. 

39 

14 

33 

4. 

45 

14.31 

4.51 

14.29 

4.57 

15 

16 

15 

30 

4. 

68 

15. 

28 

4. 

74 

15.26 

4.81 

15.24 

4.88 

16 

17 

16 

26 

4. 

97 

16 

24 

5. 

04 

16.21 

5.11 

16.19 

5.18 

17 

18 

17 

21 

5. 

26 

17 

19 

5. 

34 

17.17 

5.41 

17.14 

5.49 

18 

19 

18 

17 

5. 

56 

18 

15 

6 

63 

18.12 

5.71 

18.10 

5.79 

19 

20 

19 

13 

5. 

85 

19 

10 

5. 

93 

19.07 

6.01 

19.05 

6.10 

20 

21 

20 

08 

6. 

14 

20 

06 

6. 

23 

20.03 

6.31 

20.00 

6.40 

21 

22 

21 

04 

6 

43 

21 

01 

e 

52 

20.98 

6.62 

20.95 

6.71 

22 

23 

21 

99 

6. 

72 

21 

97 

6. 

82 

21.94 

6.92 

21.91 

7.01 

23 

24 

22. 

,95 

7. 

,02 

22, 

.92 

7. 

12 

22.89 

7.22 

22.86 

7.32 

24 

25 

23. 

,91 

7. 

31 

23, 

.88 

7 

41 

23.84 

7.52 

23.81 

7.62 

25 

26 

24. 

,86 

7. 

,60 

24 

.83 

7. 

.71 

24.80 

7.82 

24.76 

7.93 

26 

27 

25. 

,82 

7. 

,89 

25, 

.79 

8. 

,01 

25.75 

8.12 

25.71 

8.23 

27 

28 

26, 

,78 

8, 

.19 

26 

.74 

8, 

.30 

26- 70 

8.42 

26.67 

8.54 

28 

29 

27. 

.73 

8. 

,48 

27, 

.70 

8, 

,60 

27.66 

8.72 

27.62 

8.84 

29 

30 

28 

.69 

8, 

,77 

28, 

.65 

8, 

,90 

28.61 

9.02 

28.57 

9.15 

30 

31 

29, 

.65 

9, 

.06 

29 

.61 

9, 

,19 

29.57 

9.32 

29.52 

9.45 

31 

32 

30, 

.60 

! 9, 

.36 

30 

.56 

9, 

.49 

30.52 

9.62 

30.48 

9.76 

32 

33 

31 

.56 

9. 

.65 

31 

.52 

9. 

.79 

31.47 

9.92 

31.43 

10.06 

33 

34 

! 32, 

.51 

! 9, 

.94 

32 

.47 

10, 

.08 

32.43 

10.22 

32.38 

10.37 

34 

35 

33 

.47 

10, 

.23 

33 

.43 

10, 

.38 

33.38 

10.52 

33.33 

10.67 

35 

36 

34 

.43 

10. 

.53 

34 

.38 

10 

.68 

34.33 

10.83 

34.29 

10.98 

36 

37 

35 

.38 

10, 

.82 

35 

.34 

10, 

.97 

35.29 

11.13 

35  24 

11.28 

37 

38 

36 

.34 

11. 

.11 

36 

.29 

11 

.27 

36.24 

11.43 

36.19 

11.58 

38 

39 

37 

.30 

11 

.40 

37 

.25 

11 

.57 

37.19 

11.73 

37.14 

11.89 

39 

40 

38 

.25 

11 

.69 

38 

.20 

11 

.86 

38.15 

12.03 

38.10 

12.19 

40 

'41 

39 

.21 

111 

.99 

39 

.16 

12 

.16 

39.10 

12.33 

39.05 

12.50 

' 41 

42 

40 

.16 

12 

.28 

40 

.11 

12 

.45 

40.06 

12.63 

40.00 

12.80 

42 

43 

41 

. 12 

12 

.57 

41 

.07 

12 

.75 

41.01 

12.93 

40.95 

13  11 

43 

44 

42 

.08 

12 

.86 

42 

.02 

13 

.05 

41.96 

13.23 

41.91 

13  11 

44 

45 

1 43 

• 03 

13 

.16 

42 

.98 

13 

.34 

42.92 

13.53 

42.86 

13  72 

45 

46 

l 13 

.99 

13 

.45 

43 

.93 

13 

.64 

43.87 

13.83 

43.81 

14.02 

46 

47 

44 

.95 

13 

.74 

44 

.89 

13 

.94 

44.82 

14.13 

44.76 

; 14.33 

47 

48 

(45 

.90 

14 

.03 

45 

.84 

14 

.23 

45.78 

14.43 

45.71 

! 14  63 

1 48 

49 

46 

.86 

14 

.33 

46 

.80 

14 

.53 

46.73 

14.73 

46.67 

i 14.94 

49 

50 

47 

.82 

14 

.62 

47 

.76 

14 

.83 

47.69 

15.04 

47.62 

15.24 

60 

6 

I 

.9 

O 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

L&t. 

Dep. 

j Lat. 

i 

73  Deg. 

72*  Deg. 

7*1  Deg. 

72*  Deg. 

d 

09 

Q 

TRAVERSE  table 


107 


2 

CD 

r* 

17  Deg. 

17*  Deg. 

Vi 

Deg.  | 

17 1 Deg. 

[ 

O 

00* 

2 

a 

a 

® 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  i 

Dep. 

Lat. 

Dop. 

S3 

g 

51 

48.77 

14.91 

48.71 

16.12' 

48764 

15.34 

48.67 

~5i 

52 

49.73 

15.20 

49.66 

15.42 

49.59 

15.64 

49.52  j 

16.85 

62 

53 

50.68 

16.50 

50.62 

15.72 

50.55 

15.94 

50.48 

16.16 

53 

54 

51 .64 

15.79 

51.57 

16.01 

51.50 

16.24 

51.43  ! 

16.46 

54 

55 

52.60 

16.08 

52.53 

10.31 

52.45 

16.54 

52.38  | 

16.77 

55 

56 

53.55 

16.37 

53.48 

16.61 

53.41 

16.84 

53.33 

17.07 

56 

57 

54.51 

16.67  1 

54.44 

16.90 

54.36 

17.14 

54.29 

17.38 

57 

58 

55.47 

16.96 

55.39 

17.20 

55.32 

17.44 

55.24 

17.68 

58 

59 

56.42 

17.25 

56.35 

17.50 

56.27 

17.74 

56.10 

17.99 

59 

00 

57.38 

17.54 

57.30 

17.79 

57.22 

18.04 

57.14 

18.29 

60 

61 

58.33 

17.83 

58.26 

18.09 

58.18 

18.34 

58.10 

18.60 

61 

62 

59.29 

18.13 

59.21 

18.39 

59.13 

18.64 

59.05 

18.90 

62 

63 

60.25 

18.42 

60.17 

18.68 

60.08 

18.94 

60.00 

19.21 

63 

64 

61.20 

18.71 

61.12 

18.98 

61.04 

19.25 

60.95 

19.51 

64 

65 

62.16 

19.00 

62.08 

19.28 

61.99 

19.55 

61.91 

19.82 

65 

66 

63.12 

19.30 

63.03 

19.57 

62.95 

19.35 

62.86 

20.12 

66 

67 

64.07 

19.59 

63.99 

19.87 

63.90 

20.15 

63.81 

20.43 

67 

68 

65.03 

19.88 

64.94 

20.16 

64.85 

20.45 

64.76 

20.73 

68 

69 

65.99 

20.17 

65.90 

20.46 

65.81 

20.75 

65.72 

21.04 

69 

70 

66.94 

20.47 

66.85 

20.76 

66.76 

21.05 

66.67 

21.34 

70 

71 

67.90 

20.76 

67.81 

21.05 

67.71 

21.35 

67.62 

21 .65 

‘71 

72 

68.85 

21.05 

68.76 

21.35 

68.67 

21.05 

68.57 

21.95 

72 

73 

69.81 

21.34 

69.72 

21.65 

69.62 

21.95  1 

69.52 

22.26 

73 

74 

70.77 

21.64 

70.67 

21.94 

| 70.58 

22.25  j 

70.48 

22.56 

74 

75 

71.72 

21.93 

71.63 

22.24 

71.53 

22.55  ! 

71.43 

22.86 

75 

76 

72.68 

22.22 

72.58 

22.54 

72.48 

22.85 

72.38 

23.17 

76 

77 

73.64 

22.51 

73.54 

22.83  j 

73.44 

23.15 

73.33 

23.47 

77 

78 

74.59 

22.80 

74.49 

23.13 

74.39 

23.46 

74.29 

23.78  I 

78 

79 

75.55 

23.10 

75.45 

23.43 

75.34 

23.76 

75.24 

24.08 

79 

80 

76.50 

23.39 

76.40 

23.72 

76.30 

24.06 

76.19 

24.39 

80 

81 

77.46 

23.68 

77.36 

24.02 

77.25 

24.36 

77 .14 

24.  69 

81 

82 

78.42 

23.97 

78.31 

24.32 

78.20 

24.66 

78.10 

25.00 

82 

83 

79.37 

24.27 

79.27 

24.61 

79.16 

25.96 

79.05 

25.30 

83 

84 

80.33 

24.56 

80.22 

24.91 

80.11 

25.26 

80.00 

25.61 

84 

85 

81.29 

24.85 

81.18 

25.21 

81.07 

25.56 

80.95 

25.91 

85 

86 

82.24 

25.14 

82.13 

25.50 

| 82.02 

i 25.86 

81.91 

26.22 

86 

87 

83.20 

25.44 

83.09 

25.80 

'82.97 

26.16 

82.86 

26.52 

87 

88 

84.15 

25.73 

84.04 

26.10 

j 83.93 

26.46 

1 83.81 

26.83 

88 

89 

85.11 

26.02 

85.00 

26.39 

84.88 

26.76 

84.76 

27.13 

89 

90 

86.07 

26.31 

85.95 

26.69 

' 85 . 83 

27.06 

85.72 

27 . 44 

90 

91 

87.02 

26.61 

86.91 

26.99 

j 86.79 

27.36 

86^67 

27.74 

91 

92 

87  98 

26.90 

87.86 

27.28 

87.74 

27.66 

87.62 

28.05 

, 92 

93 

88.94 

27.19 

88.82 

27.58 

88.70 

27.97 

88 . 57 

28.35 

93 

94 

89.89 

27.48 

89.77 

| 27.87  i 

89.65 

28.27 

89.53 

28.66 

94 

95 

90.85 

27.78 

90.73 

28.17 

, 90.60 

28.57 

90.48 

28.96 

95 

96 

91  81 

28.07 

91.68 

28.47 

91.56 

28.87 

91.43 

1 29 . 27 

96 

97 

92.7/ 

28.36 

92.64 

28.76 

92.51 

29.17 

92.38 

29.57 

97 

98 

93.72 

28.65 

93.59 

29.06 

j 93.46 

29.47 

93.33 

29.88 

98 

99  : 94 . 67 

28.94 

94.55 

29.36 

1 94.42 

29.77 

94.29 

30.18 

99 

100 

95.63 

29.24 

95.50 

29.65 

1 95.37 

30  07 

1 95.24 

I 30.49 

100 

g 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

j Lat. 

® 

o 

a 

Q 

* 

73  Deg. 

72?  Deg. 

72i 

1 

Deg. 

72*  Deg. 

ci 

'm 

Q 

108 


TKAVKK.SK  TAflLK 


3 

5* 

pt 

p> 

18  Deg. 

184  Deg. 

18i 

Deg. 

18|  Deg. 

s> 

at 

3 

3 

Lat. 

Dep. 

Lat 

I Dep. 

Lat. 

Dep. 

Lat. 

bop. 

| 

l 

0.96 

0.31 

0.95 

0.31 

1 0.95 

'0.32 

0.95 

'0.32 

f i 

2 

1.90 

0.62 

1.90 

1 0.63 

1.90 

0.63 

1.89 

9.64 

a 

3 

2.85 

0.93 

2.85 

1 3.94 

2 84 

1 0.96 

2.84 

0.96 

9 

4 

3.80 

1.24 

3.80 

1.25 

3 79 

| 1.27 

3.79 

1 29 

! 4 

5 

4.76 

1.55 

4.76 

1.57 

4.74 

1.69 

4.73 

1.61 

! 5 

6 

5.71 

1.85 

5.70 

1.88 

5.69 

1 .90 

5.68 

1.93 

6 

7 

6.66 

2.16 

6.65 

2.19 

6.64 

1 2.22 

6.63 

2.25 

1 7 

8 

7.61 

2.47 

7.60 

2.51 

7.59 

2.54 

7.58 

2.67 

p 

9 

8.56 

2.78 

8.65 

2.82 

8.53 

2.86 

8.52 

2.89 

9 

10 

9.51 

3.09 

9.50 

3.13 

9.48 

3.17 

9.47 

3.21 

i0 

i i 

10.46 

3.40 

10.45 

3.44 

10.43 

3.49 

10.42 

3.54 

I'll 

12 

11.41 

3.71 

11.40 

3.76 

11.38 

| . 81 

11.36 

3.86 

| 12 

13 

12.36 

4 02 

12.35 

4.07 

12.33 

4 12 

12.31 

4.18 

13 

14 

13.31 

4 33 

13.30 

4.38 

13.28 

4.44 

| 13.26 ’ 

4.50 

i 14 

15 

14.27 

4.64 

14.25 

4.70 

14.22 

4.76 

1 14.20 

4.82 

1 15 

16 

15.22 

4.94 

15.20 

5.01 

15.17 

5.08 

1 15.15 

5.14 

l 16 

17 

16.17 

5.25 

16.14 

5.32 

16. 12 

5.39 

16.10 

5.46 

17 

18 

17.12 

5.56 

17.09 

5.64 

17.07 

5.71 

17.04 

5.79 

18 

19 

18.07 

5.87 

18.04 

5.95 

18.02 

6.03 

17.99 

6.11 

19 

20 

19.02 

6.18 

13  99 

6.26 

18.97 

6.35 

18.94 

6.43 

20 

21 

19.97 

6.49 

19.94 

6.58 

19.91 

6.66 

19.89 

6.75 

21 

22 

20.92 

6.80 

20.89 

6.89 

20.86 

6.98 

20.83 

7.07 

22 

23 

21.87 

7.11 

21.84 

7.20 

21.81 

7.30 

21.78  j 

7.39 

23 

24  ! 

22.83 

7.42 

22.79 

7.52 

22.76 

7.62 

22.731 

7 71 

24 

25 

23.78 

7.73 

23.74 

7.83 

23.71 

7.93 

23.67 

8.04 

25 

26 

24.73 

8.03 

24.69 

8.14 

24.66 

8.25  1 24.62 

8.36 

26 

27 

25.68 

8.34 

25.64 

8.46 

25.60 

8.57 

25.57 

8.68 

27 

28 

26.63 

8.65 

26.59 

8.77 

26.55 

8.88 

26.51 

9.00 

28 

29 

27.58 

8.96 

27.54 

9.08 

27.50 

9.20 

27.46 

9.32 

29 

30 

28.53 

9.27 

28.49 

9.39 

28.45 

9.52 

28.41 

9.64 

30 

31 

29.48 

9.58 

29.44 

9.71 

29.40 

9.84 

29.35 

9.96 

31 

32 

30.43 

9.89 

30.39 

10.02 

30.35 

10.15 

30.30 

10.29 

32 

33 

31.38 

10.20 

31.34 

10.33 

31.29 

10.47 

31.25 

10.61 

33 

34 

32.34 

JO. 51 

32.29 

10.65 

32.24 

10.79 

32 . 20 

10.93 

34 

35 

33.29 

10.82 

33.24 

10.96 

33.19 

11.11 

33.14 

11.25 

35 

36 

34.24 

11.12 

34.19 

11.27 

34.14 

11.42 

34.09 

11.67 

36 

37 

35  19 

11.43 

35.14 

11.59 

35.09 

11.74 

35.04 

11.89 

37 

38 

36.14 

11.74 

36.09 

11.90 

36.04 

12.06 

35.98 

12.21 

38 

39 

37.09 

12.05 

37.04 

12.21 

36.98 

12.37 

36.93 

12.54 

39 

■10 

38.04 

12.36 

37.99 

12.53 

37.93 

12.69 

37.88 

12.86 

40 

41 

38.99 

12.67 

38.94 

12.84 

38.88 

13.01 

'38.82 

13.18 

41 

42 

39.94 

12.98 

39.89 

13.15 

39.83 

13.33 

39.77 

13.50 

42 

43 

40.90 

13.29 

40.84 

13.47 

40.78 

13  64 

40.72 

13.82 

43 

44  1 

41.85 

13.60 

41.79 

13.78 

41.73 

13.96 

41 .66 

14.14 

44 

4f  42.80  I 

13.91 

42.74 

14.09 

42.67  I 

14.28 

42.61 

14.46 

46 

46 

43.75  1 

14.21 

43.69 

14.41 

43.62 

14.60 

43.56 

14.79 

46 

47 

44  .70 

14.52 

44.64 

14.72 

44.67 

14.91 

44.51 

15.11 

47 

48 

45.65  : 

14.83 

45.59 

15.03 

45.52 

16.23 

45.45 

15.43 

48 

49  . 

46.60  j 

15.14 

46.54 

15.35 

46.47 

16.55 

46.40 

15.75 

49 

50  1 

47.55 

15.45 

47.48 

15.66 

47.42 

16.87 

47.35 

16.07 

50 

© 

o 

C 1 

Dep.  j 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

c 

ci 

1 

Q 1 

1 

72  Deg. 

Reg. 

71*  Deg. 

71*  Deg 

5 

TRAVKKSK  TAKLF 


109 


o 

CD* 

18  Dog. 

184  Deg. 

13$  Deg. 

184  Deg. 

!g 
! 6 

3 

a 

p 

Lat. 

Dep. 

Lat. 

i 

Dep. 

Lat. 

| Dep. 

Lat. 

I Dep. 

5 

8 

51 

48. 50 

16.76 

48.43 

15.97 

48. "36 

|l67l8 

48.29" 

1 7 6.39 

61 

52 

49.45 

16.07 

49.38 

16.28 

49.31 

1 16.50 

49.24 

; 16  71 

52 

53 

50.41  16.38 

50.33 

16.60 

50.26 

16.82 

i 50.19 

17.04 

53 

54 

51.36 

16.69 

51.28 

16.91 

51.21 

17.13 

51.13 

17.36 

54 

55 

52.31 

17.00 

52.23 

17.22 

1 52.16 

17.45 

52.08 

; 17.68 

56 

56 

53.26 

17.30 

53.18 

17.54 

53.11 

17.77 

53.03 

! 18.00 

1 56 

57 

54.21 

17.61 

54.13 

17.85 

54.05 

18.09 

53.98 

! 18.32 

! 57 

58 

55.16 

17.92 

55.08 

18.16 

55.00 

18.40 

54.92 

18.64 

1 58 

59 

56.11 

18.23 

53.03 

18.48 

55.95 

18.72 

55.87 

18.96 

59 

GO 

57.06 

18.54 

56.98 

18.79 

56.90 

19.04 

56.82 

19.29 

60 

61 

58.01 

18  85 

57.93 

19.10 

57.85 

19.36 

57.76 

19.61 

61 

62 

58.97 

19.16 

58.88 

19.42 

58.80 

19.67 

58.71 

19.93 

62 

63 

59.92 

19.47 

59.83 

19.73 

59.74 

19.99 

: 59.66 

20.25 

63 

64 

60.87 

19.78 

60.78 

20.04 

60.69 

20.31 

60.60 

20.57 

64 

65 

61.82 

20.09 

61.73 

20.36 

61.64 

20.62 

61.55 

20.89 

65 

66 

62.77 

20.40 

62.68 

20.67 

62.59 

20.94 

62.50 

21.22 

66 

67 

63.72 

20.70 

63.63 

20.98 

63.54 

21.26 

63.44 

21.54 

67 

68 

64.67 

21.01 

64.58 

21.30 

64.49 

21.58 

1 64.39 

21.86 

68 

69 

65.62 

21.32 

65.53 

21 .61 

65.43 

21.89 

65.34 

22.18 

69 

70 

66.57 

21.63 

66.48 

21.92 

66.38 

22.21 

66.29 

22.50 

70 

71 

67.53 

21.94 

67.43 

22.23 

67.33 

22.53 

1 67.23 

22.82 

71 

72 

68.48 

22.25 

68.38 

22.55 

68.28 

22.85 

1 68.18 

23.14 

72 

73 

69.43 

22.56 

69.33 

22.86 

69.23 

23.16 

j 69.13 

23.47 

73 

74 

70.38 

22.8.' 

70.28 

23.17 

70.18 

23.48 

70.07 

23.79 

74 

75 

71.33 

23.18 

71.23 

23.49 

71.32 

23.80 

71.02 

24.11 

75 

76 

72.28 

23.49 

72.18 

23.80 

72.07 

24.12 

71.97 

24.43 

76 

77 

73  23 

' 23.79 

73.13 

24.11 

73.02 

24.43 

72.91 

24.75 

77 

78 

74.18 

24.10 

74.08 

24.43 

73.97 

24.75 

73.86 

25.07 

78 

79 

75.13 

24.41 

75.03 

24.74 

74.92 

25.07 

74.81 

25.39 

79 

80 

76.08 

24.72 

75 . 98 

25.05 

75.87 

25.38 

75.75 

25.72 

80 

81 

77.04 

25.03 

76.93 

25.37 

76.81 

25.70 

76.70 

26.04 

81 

82 

77.99 

25.34 

77.88 

25.68 

77.76 

26.02 

77.65 

26.36 

82 

83 

78.94 

25.65 

78.83 

25.99 

78.71 

26.34 

78.60 

26.68 

| S3 

84 

79.89 

25.96 

79.77 

26.31 

79.66 

26.65 

79.54 

27.00 

84 

85 

80.84 

26.27 

80.72 

26.62 

80.61 

26.97 

80.49 

27.32 

85 

86 

81.79 

26.58 

81.67 

26.93 

81.56 

27.29 

81.44 

27.64 

86 

87 

82.74 

26.88 

82.62 

27.25 

82.50 

27.61 

82.38 

27.97 

87 

88 

83.69 

27.19 

83.57 

27.56 

83.45 

27.92 

83.33 

28.29 

88 

89 

84.64 

27.50 

84.52 

27.87 

84.40 

28.24 

84.28 

28.61 

89 

90 

85.60 

27.81 

85.47 

28.18 

85.35 

28.56 

85.22 

28.93 

90 

91  ! 86.55 

28.12 

86.42 

28.50 

86.30 

28.37 

86.17 

29.25 

91 

92 

87.50 

28.43 

87.37 

28.81 

87.25 

29.19 

87.12 

29.57 

92 

93 

88.45 

28.74 

88.32 

29.12 

88.19 

29.51 

88.06 

29.89 

93 

94 

89  40 

29.05 

89.27 

29.44 

89.14 

29.83 

89.01 

30.22 

94 

95 

90  35 

29.36 

90.22 

29.75 

90.09 

30.14 

89.96 

30.54 

95 

96 

91  30 

29.67 

91.17 

30.06 

91.04 

30.46 

90.91 

30.86 

96 

97 

92.25 

29.97 

92.12 

30.38 

91.99 

30.78 

91.85 

31.18 

97 

98 

93.20 

30.28 

93.07 

30.69 

92.94 

31.10 

92.80 

31.50 

98 

9. 

94.15 

30.59 

94.02 

31 .00 

93.88 

31.41 

93.75 

31.82  ; 

99 

IOC 

95.11 

30.90 

94.97 

31.32 

94.83 

31.73 

194.69 

32  14  1 

.00 

I| 
« | 
b 1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat  ^ 

©' 

0 

a 

72  Deg. 

71|  Deg. 

71*  Dog. 

7H  Deg.  ! 

a) 

a 

Q 

110 


TRAVERSE  TAKLK 


c 

B 

£ 

19  Deg. 

194  Deg. 

19*  Deg. 

191  Dog. 

0 

5' 

6 

8 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

8 

e 95 

0 

33 

1)794 

0.317 

'll 794“ 

0 

.33 

“1)794 

0.34 

*1 

i 

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0 

65 

1.89 

0.66 

1.89 

0 

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1.88 

0.68 

2 

o 

2.84 

0 

98 

1 2.83 

0.99 

2.83 

I 

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2 82 

1 1.01 

3 

4 

3.78 

1 

30 

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1.32 

3.77 

1 

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3.76 

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4 

6 

4.73 

1 

63 

4.72 

l . 65 

4.71 

1 

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4.71 

1.69 

fi 

6 

5.67 

1 

95 

5.66 

1.98 

5.66 

2 

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5.65 

2.03 

6 

7 

6.62 

2 

28 

6.61 

2.31 

6.60 

2 

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6.59 

2.37 

7 

8 

7.56 

2 

60 

7.55 

2.64 

7.54 

2 

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7.53 

2.70 

8 

9 

8.51 

2 

93 

8.50 

2.97 

8.48 

3 

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8.47 

3.04 

a 

10 

9.46 

3 

26 

9.44 

3.30 

9.43 

3 

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9.41 

3.38 

10 

11 

10.40 

3 

58 

10.38 

3.63 

10.37 

I 3, 

“67 

10.35 

3.72 

n 

12 

11.35 

3 

91 

11.33 

3.96 

11.31 

4, 

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11.29 

4.06 

12 

13 

12.29 

4 

23 

12.27 

4.29 

12.25 

i 4, 

.34 

12  24 

4.39 

13 

14 

13.24 

4 

56 

13.22 

4.62 

13.20  , 

1 4, 

.67 

13  18 

4.73 

14 

15 

14.18 

4 

88 

14.16 

4.95 

14.14 

5 

.01 

14.12 

5.07 

15 

16 

15.13 

5 

21 

15.11 

5.28 

i 15.08 

5 

.34 

15.06 

5.41 

16 

17 

16.07 

5 

53 

16.05 

5.60 

16.02 

5, 

.67 

16  00 

5.74 

17 

18 

17.02 

5 

86 

16.99 

5.93 

16.97 

6, 

.01 

16.94 

6.08 

18 

19 

17.96 

6 

19 

17.94 

6.26 

17.91 

6, 

.34 

17.83 

6.42 

19 

20 

18.91 

6 

51 

18.88 

6.59 

18.85 

6, 

,68 

18.82  i 

6.76 

20 

21 

19.86 

6 

84 

19.83 

6.92 

19.80 

7. 

.01 

19.76 

7.10 

21 

22 

20.80 

7 

16 

20.77 

7.25 

20.74 

7, 

,34 

20.71  i 

7.43 

22 

23 

21.75 

7. 

49 

21.71 

7.58 

21.68 

7. 

,68  , 

21.65  1 

7.77 

23 

24 

22.69 

7. 

81 

22.66 

7.91 

22.62 

8. 

,01 

22.59  , 

8.11 

24 

25 

23.64 

8. 

14 

23.60 

8.24 

23.57 

8. 

.35 

23.53 

8.45 

25 

26 

24.58 

8. 

46 

24.55 

8.57 

24.51 

8. 

,68 

24.47  1 

8.79 

26 

27 

25.53 

8. 

,79 

25.49 

8.90 

25.45 

9. 

,01 

25.41 

9.12 

27 

28 

26.47 

9. 

,12 

26.43 

9.23 

26.39 

9. 

,35 

26.35 

9.46 

28 

29 

27.42 

9. 

,44 

27.38 

9.56 

27.34 

9. 

,68 

27.29 

9.80 

29 

30 

28.37 

9. 

,77 

28.32 

9.89 

28.28 

10. 

,01 

28.24 

10.14 

30 

31 

29.31 

10. 

09 

29.27 

10.22 

29.22 

10, 

,35 

29.18 

10  48 

31 

32 

30.26 

10. 

,42 

30.21 

10.55 

30.16 

10. 

,68 

30.12 

10  81 

32 

33 

31.20 

10. 

74 

31.15 

10.88 

31.11 

11. 

02 

31.06 

11  15 

33 

34 

32.15 

11. 

07 

32.10 

11.21 

32.05 

11. 

35 

32.00 

11  49 

34 

35 

33.09 

11. 

39 

33.04 

11.54 

32.99 

11. 

,68 

32.94 

11.83 

35 

36 

34.04 

11. 

72 

33.99 

11.87 

33.94 

12. 

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33.88 

12.17 

36 

37 

34.98 

12. 

05 

34.93 

12.20 

34.88 

12. 

,35 

34.82 

12.50 

37 

38 

35.93 

12. 

37 

35.88 

12.53 

35.82 

12. 

,68 

35.76 

12.84 

38 

39 

36.88 

12. 

70 

36.82 

12.86 

36.76 

13. 

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36.71 

13.18 

39 

40 

37  82 

13. 

02 

37. 7& 

13.19 

37.71 

13. 

35 

37.65 

13.52 

40 

41 

38.77 

13. 

35 

38.71 

13.52 

38.65 

13. 

69 

38.59 

13.85 

41 

12 

39  71 

13. 

67 

39.65 

13.85 

39.59 

14. 

,02 

39.53 

14.19 

42 

43  | 

40.66 

14. 

00 

40.60 

14.18 

40.53 

14. 

35 

40.47 

14.53 

43 

’ 14  ! 

41.60 

14. 

32 

41.54 

14.51 

41.48 

14. 

69 

41.41 

14  87 

44 

45 

42.55 

14. 

65 

42.48 

14.84 

42.42 

15. 

,02 

42.35 

15.21 

45 

16 

43.49 

14. 

98 

43.43 

15.17 

43.36 

15, 

36 

43.29 

15.54 

46 

47  j 

44.14  J 

15. 

30 

44.37 

15.50 

44.30 

15. 

69 

44.24 

15.88 

47 

18 

45.38  ! 

16. 

63 

45.32 

15.83 

45.25 

16. 

02 

45.18 

16.22 

48 

49 

46.33  1 

i5. 

96 

46.26 

16.15 

46.19 

16. 

36 

46.12 

16.50 

49 

50  , 

47.28  1 

10. 

28 

47.20 

16.48 

47.13 

16. 

,69 

47.06 

16.90 

60 

8 

a 

eS 

4-» 

on 

3 

Dtp.  j 

Lat. 

Dep.  | 

Lat. 

Dop. 

Lut. 

Dep. 

Lai. 

J Distance. 

71  Deg. 

70?  Deg. 

70, V Dog. 

70?  Deg. 

e 

e 

I 

l 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

0C 

8 

§ 

cc 

Q 


TRAVERSE  TAREK. 


Ill 


19  Deg. 

19*  DeS. 

19* 

Deg. 

19f  Deg. 

0 

5‘ 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  1 

Dep. 

a 

® 

48  22 

16.60 

48.15 

16.81 

48707 

17702“ 

48  TOO" 

17.23 

51 

49.17 

16.93 

49.09 

17.14 

49.02 

17.36 

48.94  ! 

17  .57 

52 

50  11 

17.26 

50.04 

17.47 

49.96 

17.69 

49.88  i 

17  91 

53 

51.06 

17.58 

50.98 

17.80 

50.90 

18.03 

50.82 

18  25 

54 

52.00 

17.91 

51.92 

18.13 

51.85 

18.36 

51.76 

18.59 

55 

52.95 

18  23 

52.87 

18.46 

52.79 

18.69 

52.71 

18.92 

56 

53.89 

18.56 

53.81 

18.79 

53.73 

19.03 

53.65 

19.26 

67 

54.84 

18.88 

54.76 

19.12 

54.67 

19.36 

54.59 

19.60 

58 

55.79 

19.21 

55.70 

19.45 

55.62 

19.69 

55.53 

19.94 

59 

56.73 

19.53 

56.65 

19.78 

56 . 56 

20.03 

56.47 

20.27 

60 

57.68 

19.86 

57.59 

20.11 

57.50 

20.36 

57.41 

20.61 

61 

58.62 

20.19 

58.53 

20.44 

58.44 

20.70 

58.35 

20,95 

62 

59.57 

20.51 

59.48 

20.77 

59.39 

21.03 

59.29 

21.29 

63 

60.51 

20.84 

60.42 

21.10 

60.33 

21.36 

60.24 

21.63 

64 

61.46 

21.16 

61.37 

21.43 

61.27 

21.70 

61.18 

21.96 

65 

62.40 

21.49 

62.31 

21.76 

62.21 

22.03 

62.12 

22.30 

66 

63.35 

21.81 

63.25 

22.09 

63.16 

22.37 

63.06 

22.64 

67 

64.30 

22.14 

64.20 

22.42 

64.10 

22.70 

64.00 

22.98 

68 

65.24 

22.46 

65.14 

22.75 

65.04 

23.03 

64.94 

23.32 

69 

66.19 

22.79 

66.09 

23.08 

65.98 

23.37 

65.88 

23.65 

70 

67.13 

23.12 

67.03 

23.41 

66.93 

23.70 

66.82 

23.99 

71 

68.08 

23.44 

67.97 

23.74 

67.87 

24.03 

67.76 

24.33 

72 

69.02 

23.77 

68.92 

24.07 

68.81 

24.37 

68.71 

24.67 

73 

69.97 

24.09 

69.86 

24.40 

69.76 

24.70 

69.65 

25.01 

74 

70.91 

24.42 

70.81 

24.73 

70.70 

25.04 

70.59 

25.34 

75 

71.86 

24.74 

71.75 

25.06 

71.64 

25.37 

71.53 

25.68 

76 

72.80 

25.07 

72.69 

25.39 

72.58 

25.70 

72.47 

26.02 

77 

73.75 

25.39 

73.64 

25.72 

73.53 

26.04 

73.41 

26.36 

78 

74.70 

25.72 

74.58 

26.05 

74.47 

26.37 

74.35 

26.70 

79 

75.64 

26.05 

75.53 

26.38 

75.41 

26.70 

75.29 

27.03 

80 

76.59 

26.37 

76.47 

26.70 

76.35 

27.04 

76.24 

27.37 

81 

77.53 

26.70 

77.42 

27.03 

77.30 

27.37 

77.18 

27.71 

82 

78.48 

27.02 

78.36 

27.36 

78.24 

27.71 

78.12 

28.05 

83 

79.42 

27.35 

79.30 

27.69 

79.18 

28.04 

79.06 

28.39 

84 

80.37 

27.67 

80.25 

28.02 

80.12 

28.37 

80.00 

28.72 

85 

81.31 

28.00 

81.19 

28.35 

81.07 

28.71 

80.94 

29.06 

86 

82.26 

28.32 

82.14 

28.68 

82.01 

29.04 

81.88 

29.40 

87 

83.21 

28.65 

83.08 

29.01 

92.95 

29.37 

82.82 

29.74 

88 

84.15 

28.98 

84.02 

29.34 

83.90 

29.71 

83.76 

30.07 

89 

85.10 

29.30 

84.97 

29.67 

84.84 

30.04 

84.71 

30.41 

90 

86.04 

'29.63 

85.91 

30.00 

85.78 

30.38 

85.65 

30.75 

91 

86.99 

29.95 

86.86 

30.33 

86.72 

30.71 

86.59 

31.09 

92 

87.93 

30.28 

87.80 

30.66 

87.67 

31  .04 

87.53 

31.43 

93 

88.88 

1 30.60 

88.74 

30.99 

88.61 

31.38 

88.47 

31.76 

94 

89.82  ! 30.93 

89.69 

31.32 

89.55 

31.71 

89.41 

32.10 

95 

90.77 

31.25 

90.63 

31.65 

90.49 

32.05 

90.35 

32.44 

96 

91.72 

31.58 

91.58 

31.98 

91.44 

32.38 

91.29 

32.78 

97 

92.66 

31.91 

92.52 

I 32.31 

92.38 

32.71 

92.24 

33.12 

98 

93.61 

32.23 

93.46 

1 32.64 

93.32 

33.05 

93.18 

33.45 

99 

94.65 

32.56 

94.41 

| 32.97 

94.26 

33.38 

94.12 

33.79 

100 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

a 

71  Deg. 

70f  Deg. 

70$ 

Deg. 

70*  Dog. 

2 

02 

Q 

112 


TKAVKKSE  TABLE. 


D 

s 

6 

8 

20  Deg. 

20i  Deg. 

20$ 

Deg. 

20|  Dog. 

O 

5T 

| 

.8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

l 

0.94 

0.34 

0.94 

0.36 

0.94 

0.35 

0.94 

0.31 

i 

2 

1.88 

0.68 

1.88 

0.69 

1.87 

0.70 

1.87 

0.71 

a 

3 

2.82 

1.03 

2.81 

1.04 

2.81 

1.05 

2.81 

1.06 

3 

4 

3.76 

1.37 

3.75 

1.38 

3.75 

1.40 

3.74 

1.42 

i 

6 

4.70 

1.71 

4.69 

1.73 

4.68 

1.75 

4.68 

1 1.77 

5 

6 

5.64 

2.05 

6.63 

2.08 

6.62 

2.10 

5.61 

2.13 

8 

7 

6.58 

2.39 

6.57 

2.42 

6.66 

2.45 

6.55  i 

2 48 

7 

8 

7.52 

2.74 

7.61 

2.77 

7.49 

2.80 

7.48  J 

2.83 

8 

9 

8.46 

3.08 

8.44 

3.12 

8.43 

3.15 

8.42 

3.19 

9 

10 

9.40 

3.42 

9.38 

3.46 

9.37 

3.50 

9.35 

3.54 

i0 

11 

10.34 

3.76 

10.32 

3.81 

10.30 

3.85 

10.29  i 

3.90 

11 

12 

11.28 

4.10 

11.26 

4.15 

11.24 

4.20 

11.22 

4.25 

12 

13 

12.22 

4.45 

12.20 

4.50 

12.18 

4.55 

12. 16 

4.61 

13 

14 

13.16 

4.79 

13.13 

4.86 

13.11 

4.90 

13.09 

4.96 

14 

15 

14.10 

6.13 

14.07 

5.19 

14.05 

5.25 

14.03 

5.31 

15 

16 

15.04 

5.47 

15.01. 

6.54 

14.99 

5.60 

14.96 

5.67 

16 

17 

15.97 

5.81 

15.95 

5.88 

15.92 

5.95 

15.90 

6.02 

17 

18 

16.91 

6.16 

16.89 

6.23 

16.86 

6.30 

16.83 

6.38 

18 

19 

17.85 

6.50 

17.83 

6.68 

17.80 

6.65 

17.77 

6.73 

19 

20 

18.79 

6.84 

18.76 

6.92 

18.73 

7.00 

18.70 

7.09 

20 

21 

19.73 

7.18 

19.70 

7.27 

19.67 

7.35 

19.64 

7.44 

21 

22 

20.67 

7.52 

20.64 

7.61 

20.61 

7.70 

20.57 

7.79 

22 

23 

21.61 

7.87 

21.68 

7.96 

21.54 

8.05 

21.51 

8.15 

23 

24 

22.55 

8.21 

22.52 

8.31 

22.48 

8.40 

22.44 

8.50 

24 

25 

23.49 

8.55 

23.45 

8.65 

23.42 

8.76 

23.38 

8.86 

25 

26 

24.43 

8.89 

24.39 

9.00 

24.35 

9.11 

24.31 

9.21 

26 

27 

25.37 

9.23 

25.33 

9.35 

25.29 

9.46 

25.25 

9.57 

27 

28 

26.31 

9.58 

26.27 

9.69 

26.23 

9.8i 

26.18 

9.92 

28 

29 

27.25 

9.92 

27.21 

10.04 

27.16 

10.16 

27.12 

10.27 

29 

30 

28.19 

10.26 

28.15 

10.38 

28.10 

10.51 

28.05 

10.63 

30 

31 

29.13 

10.60 

29.08 

10.73 

29.04 

10.86 

28.99 

10.98 

31 

32 

30.07 

10.94 

30.02 

11.08 

29.97 

11.21 

29.92 

11.34 

32 

33 

31.01 

11.29 

30.96 

11.42 

30.91 

11.56 

30.86 

11.69 

33 

34 

31.95 

11.63 

31.90 

11.77 

31.85 

11.91 

31.79 

12.05 

34 

35 

32.89 

11.97 

32.84 

12  11 

32.78 

12.26 

32.73 

12.40 

35 

36 

33.83 

12.31 

33.77 

12.46 

33.72 

12.61 

33.66 

12.75 

36 

37 

34.77 

12.65 

34.71 

12.81 

34.66 

12.96 

34.60 

13.11 

37 

38 

35.71 

13.00 

35.65 

13.15 

35.59 

13.31 

35.54 

13.46 

38 

39 

36.65 

13.34 

36.59 

13.50 

36.53 

13.66 

36.47 

13.82 

39 

40 

37.69 

13.68 

37.53 

13.84 

37.47 

14.01 

37.41 

14.17 

40 

'41 

38.53 

14.02 

38.47 

14.19 

38.40 

14.36 

38.34 

14.53 

41 

42 

39.47 

14.36 

39.40 

14.54 

39.34 

14.71 

39.28 

14.88 

42 

43 

40.41 

14.71 

40.34 

14.88 

40.28 

15.06 

40.21 

15.23 

43. 

44 

41.35 

16.05 

41.28 

15.23 

41.21 

15.41 

41.15 

15.59 

44  * 

45 

42.29 

15.39 

42.22 

15.58 

42.15 

15.76 

42.08 

15.94  1 

45  * 

4e 

43.23 

15.73 

43.16 

15.92 

43.09 

16.11 

43.02 

16  30  1 

46 

4" 

44.17 

16.07 

44.09 

16.27 

44.02 

16.46 

43.95 

16  65  | 

47 

48 

45.  ii 

16.42 

46.03 

16.61 

44.96 

16  81 

44.89 

17.01 

48 

49 

46.04 

16.76 

45.97 

16.96 

45.90 

17.16 

45.82 

17.36 

49 

60 

46.98 

■ 17.10 

46.91 

17.31 

46.83 

17.61 

46.76 

17.71 

50 

8 

§ 

.2 

Q 

Dep. 

1 Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

a> 

o 

e 

$ 

m 

a 

70  Deg. 

69J  Deg. 

G9J  Deg. 

69*  Dog. 

TRAVERSE  TABLE. 


113 


1 

q 

20  Deg. 

20*  De?. 

2QA  Deg 

20|  Deg. 

3 

2- 

a 

o 

® 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 

8 

51 

47 .92i 

17.44 

47T5 

17.65 

47777 

17.86 

47.69 

18.07 

51 

52 

48.86 

17.79 

48.79 

18.00 

48.71 

18.21 

48.63 

18.42 

52 

53 

49.80 

18.13 

49.72 

18.34 

49.64 

18.56 

49.56 

18.78 

53 

54 

50.74 

18.47 

50.66  ! 

18.69 

50.58 

18.91 

50.50 

19.13 

54 

55 

51.68 

18.81 

51.60 

19.04 

51.52 

19.26 

51.43 

19.49 

55 

56 

52.62 

19.15 

52.54 

19.38 

52.45 

19.61 

52.37 

19.84 

56 

57 

53.56 

19.50 

53.48 

19.73 

53.39 

19.96 

53.30 

20.19 

57 

58 

54.50 

19.84 

54.42 

20.07 

54.33 

20.31 

54.24 

20.55 

58 

59 

55.44 

20.18 

55.35 

20.42 

55.26 

20.66 

55.17 

20.90 

59 

60 

56.38 

20.52 

56.29 

20.77 

56.20 

21.01 

56.11 

21.26 

60 

61 

57-32 

20.86 

57.23 

21.11 

57.14 

21.36 

57.04 

21.61 

'61 

62 

58.26 

21.21 

58.17 

21.46 

58.07 

21.71 

57.98 

21.97 

62 

63 

59.20 

21.55 

59.11 

21.81 

59.01 

22.06 

58.91 

22.32 

63 

64 

60.14 

21.89 

60.04 

22.15 

59.95 

22.41 

59.85 

22.67 

64 

65 

61.08 

22.23 

60.98 

22.50 

60.88 

22.76 

60.78 

23.03 

65 

66 

62.02 

22.57 

61.92 

22.84 

61.82 

23.11 

61.72 

23.38 

66 

67 

62.96 

22.92 

62.86 

23.19 

62.76 

23.46 

62.65 

23.74 

67 

68 

63.90 

23.26 

63.80 

23.54 

63.69 

23.81 

63.59 

24.09 

68 

69 

64.84 

23.60 

64.74 

23.88 

6t.63 

24.16 

64.52 

24.45 

69 

70 

65.78 

23.94 

65.67 

24.23 

65.57 

24.51 

65.46 

24.80 

70 

71 

66.72 

24.28 

66.61 

24.57 

66.50 

24.86 

66.39 

25.15 

71 

72 

67.66 

24.63 

67.55 

24.92 

67.44 

25.21 

67.33 

25.51 

. 72 

73 

68.60 

24.97 

68.49 

25.27 

68.38 

25.57 

68.26 

25.86 

73 

74 

69.54 

25.31 

69,43 

25.61 

69.31 

25.92 

69.20 

26.22 

74 

75 

70.48 

25.65 

70.36 

25.96 

70.25 

26.27 

70.14 

26.57 

75 

76 

71.42 

25.99 

71.30 

26.30 

71.19 

26.62 

71.07 

26.93 

76 

77 

72.36 

26.34 

72.24 

26.65 

72.12 

26.97 

72.01 

27.28 

77 

78 

73.30 

26.68 

73.18 

27.00 

73.06 

27.32 

72.94 

27.63 

78 

79 

74.24 

27.02 

74.12 

27.34  ! 

74.00 

27.67 

73.88 

27.99 

79 

80 

, 75.18 

27.36 

75.06 

27.69  1 

74.93 

28.02 

74.81 

28.34 

80 

81 

76.12 

27.70 

75.99 

28.04  ! 

75.87 

28.37 

75.75 

28.70 

81 

82 

77.05 

28.05 

76.93 

28.38  j 

76.81 

28.72 

76.68 

29.05 

82 

83 

77.99 

28.39 

77.87 

28.73  i 

77.74 

29.07 

77.62 

29.41 

83 

84 

78.93 

28.73 

78.81 

29.07 

78.68 

29.42 

78.55 

29.76 

84 

85 

79.87 

29.07 

79.75 

29.42 

79.62 

29.77 

79.49 

30.11 

85 

86 

80.81 

29.41 

80.68 

29.77 

80.55 

30.12 

80.42 

30.47 

86 

87 

81.75 

29.76 

81.62 

30.11 

81.49 

30.47 

81.36 

30.82 

87 

88 

1 82.69 

30.10 

82.56 

30.46 

82.43 

30.82 

82.29 

31.18 

88 

89 

| 83.63 

30.44 

83.50 

30.80 

83.36 

31.17 

83.23 

31.53 

89 

90 

| 84.57 

30.78 

84.44 

31.15 

84.30 

31.52 

84.16 

31.89 

90 

91 

1 85.51 

| 31.12 

85.38 

31.50 

85.24 

31.87 

85.10 

32  24 

91 

92 

1 86.45 

1 31.47 

86.31 

31.84 

86.17 

32.22 

86.03 

32.59 

92 

93 

87.39 

31.81 

87.25 

32.19 

87. K 

32.57 

86.97 

32.95 

93 

94 

88.33 

32.15 

88.19 

32.54 

88.05 

32.92 

87.90 

33.30 

94 

95 

89.27 

32.49 

89.13 

32.88 

88.98 

33.27 

88.84 

33.66 

95 

96 

90.21 

32.83 

90.07 

33.23 

89.92 

33.62 

89.77 

34.01 

96 

97 

1 91.15 

: 33.18 

91.00 

33.57 

90.86 

33.97 

90.71 

34.37 

97 

98 

92.09 

33.52 

91.94 

33.92 

91.79 

34.32 

91.64 

34.72 

98 

99 

93.0.1 

33.86 

92.88 

34.27 

92.73 

34.67 

92.58 

35.07 

99 

100 

| 93.97 

! 34.20 

93.82 

34.61 

93.67 

35.02 

93.51 

35.43 

100 

f 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

CD 

O 

c 

3 

.9 

Q 

l 

1 

| 70  Deg. 

69f  Deg. 

69  i 

Deg 

69*  Dog 

n 

cci 

3 

114 


TRAVERSE  TABLE. 


3 

5' 

§ 

s 

21  Deg. 

21}  Deg. 

21 1 Deg. 

1 

21 1 Dog. 

Distance. 

Lat. 

Dep. 

Lat. 

Dop. 

L&t. 

Dep. 

Lat. 

Dep. 

1 

0 

93 

0 

36 

0 

93 

0. 

36 

0. 

03 

0.37 

0, 

.93 

0.37 

1 

2 

1 

87 

0 

72 

1 

86 

0. 

72 

1. 

,86 

0.73 

1. 

,86 

0.74 

2 

3 

2 

80 

1 

08 

2 

80 

1. 

09 

2. 

.79 

1.10 

2. 

,79 

1.11 

3 

4 

3 

73 

1 

43 

3 

73 

1. 

45 

3. 

,72 

1.47 

3. 

,72  1 

1 .48 

4 

6 

4 

67 

1 

79 

4 

66 

1. 

81 

4. 

,65 

1.83 

4, 

64 

1 .85 

5 

6 

5 

60 

2 

15 

5 

59 

2. 

17 

5. 

,58 

2.20 

6. 

,67 

2.22 

6 

7 

6 

54 

2 

51 

6 

52 

2. 

54 

6, 

,51 

2.57 

6. 

,50 

2.59 

7 

8 

7 

47 

2 

87 

7 

46 

2. 

90 

7, 

.44 

2.93 

7, 

.43 

2.96 

8 

9 

8 

40 

3 

23 

8 

39 

3. 

26 

8. 

,37 

3.30 

8. 

.36 

3.34 

9 

10 

9 

34 

3 

58 

9 

32 

3. 

62 

9. 

.30 

3.67 

9. 

,29 

3.71 

10 

11 

10 

27 

3 

94 

10 

25 

3. 

99 

10, 

.23 

”4.03 

10. 

22 

4.08 

11 

12 

11 

20 

4 

30 

11 

18 

4. 

35 

11, 

,17 

4.40 

11. 

,15  1 

4.45 

12 

13 

12 

14 

4 

66 

12 

12 

4. 

71 

12, 

.10 

4.76 

12, 

,07 

4.82 

13 

14 

13 

07 

5 

02 

13 

05 

5. 

07 

13, 

.03 

5. 13 

13, 

.00 

5.19 

14 

15 

14 

00 

5 

38 

13 

98 

5. 

44 

13, 

.96 

5.50 

13, 

.93 

5.66 

15 

1G 

14 

94 

5 

73 

14 

91 

5. 

80 

14, 

.89 

5.86 

14, 

,86 

5.93 

16 

17 

15 

87 

6 

09 

15 

84 

6. 

, 16 

15, 

.82 

6.23 

15, 

.79 

6.30 

17 

18 

16 

80 

6 

45 

16 

78 

6. 

52 

16, 

.75 

6.60 

16, 

.72 

6.67 

18 

19 

17 

74 

6 

81 

17 

71 

6. 

.89 

17, 

.68 

6.96 

17, 

.65 

7.04 

19 

20 

18 

67 

7 

17 

18 

64 

7. 

,25 

18, 

.61 

7.33 

18, 

.58 

7.41 

20 

21 

19 

61 

7 

53 

19 

57 

7. 

,61 

19, 

.54 

7.70 

19 

.50  | 

7.78 

21 

22 

20 

54 

7 

88 

20 

50 

7. 

,97 

20, 

.47 

8.06 

20, 

.43 

8.15 

22 

23 

21 , 

.47 

8. 

,24 

21, 

.44 

8. 

,34 

21, 

.40 

8.43 

21, 

.36 

8.52 

23 

24 

22, 

.41 

8. 

,60 

22 

.37 

8. 

,70 

22 

.33 

8.80 

22, 

.29 

8.89 

24 

25 

23, 

.34 

8. 

,98 

23 

.30 

9. 

,06 

23 

.26 

9.16 

23, 

.22 

9.26 

25 

26 

24, 

.27 

9, 

,32 

24, 

.23 

9, 

,42 

24, 

.19 

9.53 

24, 

.15 

9.63 

26 

27 

25 

.21 

9, 

,68 

25 

.16 

9. 

,79 

25 

.12 

9.90 

25, 

.08 

10.01 

27 

28 

26 

.14 

10, 

.03 

26 

.10 

10, 

.15 

26 

.05 

10.26 

26, 

.01 

| 10  38 

28 

29 

27 

.07 

10, 

,39 

27 

.03 

10, 

.51 

26, 

.98 

10.63 

26 

.94 

10.75 

29 

30 

28 

.01 

10, 

.75 

27, 

.96 

10, 

.87 

27, 

.91 

11.00 

27, 

.86 

1 11.12 

30 

31 

28, 

.94 

11, 

.11 

28, 

.89 

11. 

.24 

28 

.84 

11.36 

28 

.79 

11.49 

31 

32 

29 

.87 

11, 

.47 

29, 

.82 

11, 

,60 

29 

.77 

11.73 

29, 

.72 

11.86 

32 

33 

30 

.81 

11, 

.83 

30 

.76 

11, 

.96 

30, 

.70 

12.09 

30, 

.65 

12.23 

33 

34 

31 

.74 

12, 

.18 

31 

.69 

12, 

.32 

31 

.63 

12.46 

31 

.58 

12.60 

34 

35 

32 

.68 

12, 

.54 

32 

.62 

12, 

.69 

32 

.56 

12.83 

32 

.51 

12.97 

35 

36 

33 

.61 

12, 

.90 

33 

.55 

13, 

.05 

33 

.50 

13.19 

33 

.44 

13.34 

36 

37 

34 

.54 

13, 

.26 

34 

.48 

13, 

.41 

34 

.43 

13.56 

34 

.37 

13.71 

37 

38 

35 

.48 

13, 

.62 

35 

.42 

13, 

.77 

35 

.36 

13.93 

35 

.29 

14.08 

38 

39 

36 

.41 

13, 

.98 

36 

.35 

14, 

.14 

36 

.29 

14.29 

36 

.22 

14.45 

39 

40 

37 

.34 

14, 

.33 

37 

.28 

14, 

.50 

37 

.22 

14.66 

37 

.15 

14.82 

40 

41 

38 

.28 

14, 

.69 

38 

.21 

14, 

.86 

38 

.15 

15.03 

38 

.08 

15.19 

41 

42 

39 

.21 

15 

,05 

39 

. 14 

15, 

.22 

39 

.08 

15.39 

39 

.01 

15.56 

42 

43 

40 

. 14 

15 

.41 

40 

.08 

15, 

.58 

40 

.01 

15.76 

39 

.94 

15.93 

43 

44 

41 

.08 

15, 

.77 

41 

.01 

15, 

.95 

40 

.94 

16.13 

40 

.87 

16.30 

44 

45 

42 

.01 

16 

.13 

41 

.94 

16 

.31 

41 

.87 

16.49 

41 

.80 

16.68 

45 

46 

42 

.94 

16 

.48 

42 

.87 

16 

.67 

42 

.80 

16.86 

42 

.73 

17.05 

46 

47 

43 

.88 

16 

.84 

43 

.80 

17 

.03 

43 

.73 

17.23 

43 

.65 

17.42 

47 

48 

44 

.81 

17 

.20 

44 

.74 

17 

.40 

44 

.66 

17.59 

44 

.58 

17.79 

48 

49 

45 

.75 

17 

.56 

45 

.67 

17, 

.76 

45 

.59 

17.96 

45 

.51 

18.16 

49 

50 

46 

.68 

17 

.92 

46 

.60 

18, 

. 12 

46 

.52 

18.33 

46 

.44 

18.53 

50 

8 

| 

j 

a 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

1 Lat. 

Dep. 

! Lat. 

, 

© 

o 

a 

$ 

to 

5 

69  De<r. 

68}  Deg 

681 

Dog. 

58}  Dog. 

1 

TRAVERSE  TABLE. 


115 


CJ 

5 

£ 

21  Deg 

21}  Deg. 

21*  Deg. 

21 1 Deg. 

a 

s 

s 

s 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

§ 

61 

47.61 

18.28 

47.53 

18.48 

47.45 

18.69 

47.37 

78.90 

51 

62 

48.55 

18.64 

48.46 

18  85 

48.38 

19.06 

48.30 

19.27 

62 

63 

49.48 

18.99 

49.40 

19.21 

49.31 

19.42 

49.23 

19.64 

53 

64 

50.41 

19.35 

50.33 

19.57 

50.24 

19.79 

50.16 

20.01 

54 

55 

51  35 

19.71 

51.26 

19.93 

61.17 

20.16 

51.08 

20.38 

55 

66 

62  28 

20.07 

52.19 

20.30 

52.10 

20.52 

52.01 

20.75 

56 

57 

63  21 

20.43 

53.12 

20.66 

53.03 

20.89 

52.94 

21.12 

57 

58 

54.15 

20.79 

54.06 

21.02 

53.96 

21.26 

53.87 

21.49 

58 

59 

55.08 

21.14 

54.99 

21.38 

54.89 

21.62 

54.80 

21.86 

59 

60 

56.01 

21.50 

55.92 

21.75 

55.83 

21.99 

55.73 

22.23 

60 

61 

56.95 

21.86 

56.85 

22.11 

56.76 

22.36 

56.66 

22.60 

61 

62 

57  88 

22.22 

57.78 

22.47 

57.69 

22.72 

57.59 

22.97 

62 

63 

58.82 

22.58 

58.72 

22.83 

58.62 

23.09 

58.52 

23.35 

63 

64 

59.75 

22.94 

59.65 

23.20 

59.55 

23.46 

59.44 

23.72 

64 

65 

60.68 

23.29 

60.58 

23.56 

60.48 

23.82 

60.37 

24.09 

65 

66 

61.62 

23.65 

61.51 

23.92 

61.41 

24.19 

61.30 

24.46  1 

66 

67 

62.55 

24.01 

62.44 

24.28 

62.34 

24.56 

62.23 

24.83 

67 

68 

63.48 

24.37 

63.38 

24.65 

63.27 

24.92 

63.16 

25.20 

68 

69 

64.42 

24.73 

64.31 

25.01 

64.20 

25.29 

64.09 

25.57  I 

69 

70 

65.35 

25.09 

65.24 

25.37 

65.13 

25.66 

65.02 

25.94 

70 

7) 

66.28 

25.44 

66.17 

25.73 

66.06 

26.02 

65.95 

26.31 

71 

72 

67.22 

25.80 

67.10 

26.10 

66.99 

26.39 

66.87 

26.68 

72 

73 

68.16 

26.16 

68.04 

26.46 

67.92 

26.75 

67.80 

27.05 

73 

74 

69.08 

26.52 

68.97 

26.82 

68.85 

27.12 

68.73 

27.42 

74 

75 

70.02 

26.88 

69.90 

27.18 

69.78 

27.49 

69.66 

27.79 

75 

76 

70.95 

27.24 

70.83 

27.55 

70.71 

27.85 

70.59 

28.16 

76 

77 

71.89 

27.59 

71.76 

27.91 

71.64 

28.22 

71.52 

28.53 

77 

78 

72.82 

27.95 

72.70 

28.27 

72.57 

28.59 

72.45 

28.90 

78 

79 

73.75 

28.31 

73.63 

28.63 

73.50 

28.95 

73.38 

29.27 

79 

80 

74.69  | 

j 28.67 

74.56 

29.00 

74.43 

29.32 

74.30 

29.64 

80 

81 

75.62 

29.03 

75.49 

29.36 

75.36 

29.69 

75.23 

30.02 

81 

82 

76.55 

29.39 

76.42 

29.72 

76.29 

30.05 

76.16 

30.39 

82 

83 

77.49 

29.74 

77.36 

30.08 

77.22 

30.42 

77.09 

30.76 

83 

84 

78.42 

30.10 

78.29 

30.44 

78.16 

30.79 

78.02 

31.13 

84 

85 

79.35 

30.46 

79.22 

30.81 

79.09 

31.15 

78.95 

31.50 

85 

86 

80.29 

30.82 

80.15 

31.17 

80.02 

31.52 

79.88 

31.87 

86 

97 

81.22 

31.18 

81.08 

31.53 

80.95 

31.89 

80.81 

32.24 

87 

88 

82.16 

31  54 

82.02 

31.89 

81.88 

32.25 

81.74 

32.61 

88 

89 

83.09 

31.89 

82.95 

32.26 

82.81 

32.62 

82.66 

32.98 

89 

90 

84.02 

32.25 

83.88 

32.62 

83.74 

32.99 

83.59 

33.35 

90 

91 

84 . 96 

32.61 

84.81 

32.98 

”84.67 

33.35 

84.52 

33.72 

91 

92 

85 . 89 

32.97 

85.74 

33.34 

85.60 

33.72 

85.45 

34.09 

92 

93 

86.82 

33.33 

86.68 

33.71 

86.53 

34.08 

86.38 

34.46 

. 93 

94 

87.76 

33.69 

87.61 

34.07 

87.46 

34.45 

87.31 

34.83 

1 94 

95 

88.69 

34.04 

88.54 

34.43 

1 88.39 

34.82 

88.24 

35.20 

i 95 

96 

89.62 

34.40 

89.47 

34.79 

89.32 

35.18 

89.17 

35.57 

! 96 

97 

90.56 

34.76 

90.40 

35.16 

90.25 

35.55 

90.09 

35.94 

97 

98 

91.49 

35.12 

91.34 

35.52 

91.18 

35.92 

91.02 

36.31 

| 98 

99 

92.42 

35.48 

92.27 

35.88 

92.11 

36.28 

91.95 

36.69 

99 

100 

93.36 

35.84 

93.20 

36.24 

93.04 

36.65 

92.88 

37.06 

mo 

i 

c 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

'i 

C 

a 

.5 

Q 

69  Deg. 

68*  Deg. 

Oft*  Deg. 

68*  Deg 

cd 

03 

5 

22 


TRAVtKSK  TABLK. 


116 


» 

22  Deg. 

22i  Dog. 

Deg. 

22\  Dog. 

"1 

5 

o 

9 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

) 

1 

5.93 

0.37 

0.93 

0.38 

0.92 

0.38 

0.92 

0.39 

"i 

2 

1.85 

0.75 

1.85 

0.76 

1.85 

0.77 

1.84 

0.77  | 

2 

3 

2.78 

1.12 

2.78 

1.14 

2.77 

1.15 

2.77 

1.16  | 

3 

4 

3.71 

1.50 

3.70 

1.51 

3.70 

1.53 

3.69 

’ .55 

4 

5 

4.64 

1.87 

4.63 

1.89 

4.62 

1.91 

4.61 

1.93 

5 

6 

5.56 

2.25 

5.55 

2.27 

5.54 

2.30 

5.53 

2.32 

6 

7 

6.49 

2.62 

6.48 

2.65 

6.47 

2.68 

6.46 

2.71 

7 

8 

7.42 

3.00 

7.40 

3.03 

7.39 

3.06 

7 38 

3.09 

8 

9 

8.34 

3.37 

8.33 

3.41 

8.31 

3.44 

8.30 

3.48 

9 

10 

9.27 

3.75 

9.26 

3.79 

9.24 

3.83 

9.22 

3.87 

10 

11 

10.20 

4.12 

10.18 

4.17 

10.16 

‘ 4.21 

10.14 

4.25 

11 

12 

11.13 

4.50 

11.11 

4.54 

11.09 

4.59 

11.07 

4.64 

12 

13 

12.05 

4.87 

12.03 

4.92 

12.01 

4.97 

11.99 

5.03 

13 

14 

12.98 

5.24 

12.96 

5.30 

12.93 

5.36 

12.91 

5.41 

14 

15 

13.91 

5.62 

13.88 

5.68 

13.86 

•5.74 

13.83 

5.80 

15 

16 

14.83 

5.99 

14.81 

6.06 

14.78 

6.12 

14.76 

6.19 

16 

17 

15.76 

6.37 

15.73 

6.44 

15.71 

6.51 

15.68 

6.57 

17 

18 

16.69 

6.74 

16.66 

6.82 

•16.63 

6.89 

16.60 

6.96 

18 

19 

17.62 

7.12 

17.59 

7.19 

17.55 

7.27 

17.52 

7.35 

19 

20 

18.54 

7.49 

18.51 

7.57 

18.48 

7.65 

18.44 

7.73 

20 

21 

19.47 

7.87 

19.44 

7.95 

19.40 

8.04 

'19.37 

8.12 

21 

22 

20.40 

8.24 

20.36 

8.33 

20.33 

8.42 

20.29 

8.51 

22 

23 

21.33 

8.62 

21.29 

8.71 

21.25 

8.80 

21.21 

8.89 

23 

24 

22.25 

8.99 

22.21 

9.09 

22.17 

9.18 

22.13 

9.28 

24 

25 

23.18 

9.37 

23.14 

9.47  1 

23.10 

9.57  : 

23.05 

9.67 

25 

26 

24.11 

9.74 

24- 06 

9.84 

24.02 

9.95 

23.98 

10.05 

26 

27 

25.03 

10.11 

24.99 

10.22 

24.94 

10.33 

24.90 

10.44 

27 

28 

25.96 

10.49 

25.92 

10.60 

25.87 

10.72  1 

i 25.82 

10.83 

28 

29 

26.89 

10.86 

26.84 

10.98 

26.79 

11.10 

26.74 

11.21 

29 

30 

27.82 

11.24 

27.77 

11.36 

27.72 

11.48 

| 27.67 

11.60 

30 

31 

28.74 

11.61 

28.69 

11.74 

28.64 

11.86 

28.59 

11.99 

31 

32 

29.67 

11.99 

29.62 

12.12 

29.56 

12.25 

29.51 

12.37 

32 

33 

30.60 

12.36 

30.54 

12.50 

30.49 

12.63  J 

30.43 

12.76 

33 

34 

31.52 

12.74 

31.47 

12.87 

31.41 

13.01  ! 

1 31.35 

13.15 

34 

35 

32.45 

i 13.11 

32.39 

13.25 

32.34 

13.39 

32.28 

13.53 

35 

36 

33.38 

13.49 

33.32 

13.63 

33.26 

13.78 

33.20 

13.92 

36 

37 

34.31 

13.86 

34.24 

14.01 

34.18 

14.16 

34.12 

14.31 

37 

38 

35.23 

14.24 

35.17 

14.39 

35.11 

14.54 

35.04 

14.70 

38 

39 

36.16 

14.61 

36.10 

14.77 

36.03 

14.92 

35.97 

15.08 

39 

40 

37.09 

14.98 

37.02 

15.15 

36.96 

15.31 

36.89 

15.47 

40 

41 

38.01 

15.36 

37.95 

15.52 

37.88 

15-  69 

37.81 

15.36 

41 

42 

38.94 

15.73 

38.87 

15.90 

38.80 

16.07 

38.73 

10.24 

42 

43 

39.87 

16.11 

39.80 

16.28 

39.73 

16.46 

39.65 

16.63 

43 

44 

40.80 

16.48 

40.72 

16.66 

40.65 

16.84 

40.58 

17.02 

44 

45 

41.72 

16.86 

41.65 

17.04 

41.57 

17.22 

41.50 

17  .40 

45 

46 

42.65 

17.23 

42.57 

17.42 

12.50 

17.60 

42.42 

17.79 

i 46 

47 

43.58 

17.61 

43.50 

17.80 

43.42 

17.99 

43.34 

18.18 

47 

48 

i 44.50 

17.98 

44.43 

18.18 

44.35 

18.37 

44.27 

18.56 

48 

49 

45.43 

18.36 

45.35 

18.55 

45.27 

18.75 

45.19 

18.95 

49 

50 

46.36 

18.73 

46.28 

18.93 

46.19 

19.13 

46.11 

19.34 

60 

8 

Dop. 

| Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

! § 

Q 

68  Deg. 

67  j Dog. 

67  j Deg. 

67*  Deg. 

1 

1 s 

1 ® 

i 

TRAVERSE  TABLE. 


117 


O i 
S'  1 

22  Dog. 

224  Dog. 

22*  Deg. 

m Deg. 

a 

S' 

3 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| 

51  1 47.29 

19.10 

47.20 

19.31 

47.12 

19.52 

47703 

19.72 

51 

52 

48.21 

19.48 

48.13 

19.69 

48.04 

19.90 

47.95 

20.11  | 

62 

53 

49.14 

19.85 

49.05 

20.07 

48.97 

20.28 

48.88 

20.50  | 

53 

64 

50.07 

20.23 

49.98 

20.45 

49.89 

20.66 

49.80 

20.88 

54 

55 

51.00 

20.60 

50.90 

20.83 

50.81 

21.05 

50.72 

21.27 

55 

56 

51.92  i 20.98 

51.83 

21.20 

51.74 

21.43 

51.64 

21.66  ! 

56 

57 

52.85 

21.35 

52.76 

21.58 

52.66 

21.81 

52.57 

22.04 

57 

58 

53.78 

21.73 

53.68 

21.96 

53.59 

22.20 

53.49 

22.43 

58 

59 

54.70 

22.10 

54.61 

22.34 

54.51 

22.58 

54.41 

22.82 

59 

60 

55.63 

22.48 

55.53 

22.72 

55.43 

22.96 

55.33 

23.20 

60 

61 

56.56 

22.85 

56.47 

23.10 

56.36 

23.34 

56.25 

23.59 

61 

62 

57.49 

23.23 

57.38 

23.48 

57.28 

23.73 

57.18 

23.98  | 

62 

6.1 

58.41 

23.60 

58.31 

23.85 

58.20 

24.11 

58.10 

24.36 

63 

64 

59.34 

23.97 

59.23 

24.23 

59.13 

24.49 

59.02 

24.75 

64 

65 

60.27 

24.35 

60.16 

24.61 

60.05 

24.87 

59.94 

25.14 

65 

66 

61.19 

24.72 

61.09 

24.99 

60.98 

25.26 

60.87 

25.52 

66 

67 

62.12 

25.10 

62.01 

25.37 

61.90 

25.64 

61.79 

25.91 

67 

68 

63.05 

25.47 

62.94 

25.75 

62.82 

26.02 

62.71 

26.30 

68 

69 

63.98 

25.85 

63.86 

26.13 

63.75 

26.41 

63.63 

26.68 

69 

70 

64.90 

26.22 

64.79 

26.51 

64.67 

26.79 

64.55 

27.07 

70 

71 

65.83 

26.60 

65.71 

26.88 

65.60 

27.17 

65.48 

27.46 

71 

72 

66.76 

26.97 

66.64 

27.26 

66  52 

27.55 

66.40 

27.84 

72 

73 

! 67.68 

27.35 

67.56 

27.64 

67.44 

27.94 

67.32 

28.23 

73 

74 

68.61 

27.72 

68.49 

28.02 

68.37 

28.32 

68.24 

28.62 

74 

75 

69.54 

28.10 

69.42 

28.40 

69.29 

28.70 

69.17 

29.00 

75 

76 

70.47 

28.47 

70.34 

28.78 

70.21 

29.08 

70.09 

29.39 

76 

77 

71.39 

28.84 

71.27 

29.16 

71.14 

29.47 

71.01 

29.78 

77 

78 

72.32 

29.22 

72.19 

29.53 

72.06 

29.85 

71.93 

30.16 

78 

79 

73.25 

29.59 

73.12 

29.91 

72.99 

30.23 

72.85 

30.55 

79 

80 

74.17 

29.97 

74.04 

30.29 

73.91 

30.61 

73.78 

30.94 

80 

81 

75.10 

30.34 

74.97 

30.67 

74.83 

31 .00 

74.70 

31.32 

81 

82 

76.03 

30.72 

75.89 

31.05 

75.76 

31 .38 

75.62 

31.71 

82 

83 

76.96 

31.09 

76.82 

31.43 

76.68 

31.76 

76.54 

32.10 

83 

84 

77.88 

31.47 

77.75 

31 .81 

77.61 

32.15 

77.46 

32.48 

84 

85 

78.81 

31.84 

78.67 

32.19 

78.53 

32.53 

78.39 

32.87 

85 

86 

79.74 

32.22 

79.60 

32.56 

79.45 

32.91 

79.31 

33.26 

86 

87 

80.66 

32.59 

80.52 

32.94 

80.38 

33.29 

80.23 

33.64 

87 

88 

81.59 

32.97 

81.45 

33.32 

81.30 

33.68 

81.15 

34.03 

88 

89 

82.52 

33.34 

82.37 

33.70 

82.23 

34.06 

82.08 

34.42 

89 

90 

| 83.45 

33.71 

83.30 

34.08 

83.15 

34.44 

83.00 

34.80 

90 

91 

84.37 

34.09 

84.22 

34.46 

84.07 

34.82 

83.92 

35.19 

91 

92 

85.30 

34.46 

85.15 

34.84 

85.00 

35.21 

84.84 

35.58 

92 

93 

86.23 

34.84 

86.08 

35.21 

85.92 

35.59 

85.76 

35.96 

93 

94 

87.16 

35.21 

87.00 

35.59 

86.84 

35.97 

86.69 

36.35 

94 

95 

88  08 

| 35.59 

87.93 

35.97 

87.77 

36.35 

87.61 

36.74 

; 9£ 

96 

89.01 

! 35.96 

88.85 

36  35 

88.69 

36.74 

88.53 

37.12 

1 96 

9" 

89.94 

36.34 

89.78 

36.73 

89.62 

37.12 

89.46 

37  51 

97 

96 

90  86 

36.71 

90.70 

37.11 

90.54 

i 37.50 

90.38 

37.90 

l 98 

96 

91.79 

37.09 

91.63 

37.49 

91.46 

37.89 

91.30 

38.28 

! 99 

(00 

92  72 

37.46 

92.55 

! 37.86 

192.39 

38.27 

92.22 

38.67 

100 

© 

o 

§ 

GO 

j Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

« 

© 

§ 

*-» 

00 

5 

08  Deg. 

67f  Deg. 

' 

67*  Deg. 

07*  Deg. 

Q 

118 


r KAVKfiSK  TAHLE. 


D 

r 

£ 

23  Deg. 

234  Deg. 

23*  Deg. 

23|  Deg. 

5’ 

£ 

6 

2 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat.  | 

Dep. 

p 

8 

l 

0.92 

0.39 

0.92 

0.39 

0 . 92 

'M0 

0.92 

' MO 

I 

2 

1.84 

0.78 

1 .84 

0.79 

1.83 

0.80 

1.83 

0.81 

2 

3 

2.76 

1.17 

2.76 

1.18 

2.75 

1.20 

2.76 

1.21 

3 

4 

3.68 

1.56 

3.68 

1.58 

3.67 

1.59 

3.66 

1.61 

1 

6 

4.60 

1.95 

4.59 

1.97 

4.69 

1.99 

4.58 

2.01 

6 

5.52 

2.34 

5.51 

2.37 

5.50 

2.39 

5.49 

2.42 

6 

7 

6.44 

2 74 

6.43 

2.76 

6.42 

2.79 

6.41 

2.82 

7 

8 

7.36 

3.13 

7.35 

3.16 

7.34 

3.19 

7.32 

3.22 

8 

9 

8.28 

3.52 

8.27 

3.55 

8.25 

3.59 

8.24 

3.62 

9 

10 

9.20 

3.91 

9.19 

3.95 

9.17 

3.99 

9.15 

4.03 

0 

11 

10.13 

4.30  | 

10.11 

4.34 

10.09 

4.39 

10.  OT 

4.43 

11 

12 

11.05 

4.69 

11.03 

4.74 

11.00 

4.78 

10.98  | 

4.83 

12 

13 

11.97 

5.08 

11.94 

5.13 

11.92 

5.18 

11.90 

6.24 

13 

14 

12.89 

5.47 

12.86 

5.53 

12.84 

5.58 

12.81  1 

5.64 

14 

15 

13.81 

5.86 

13.78 

5.92 

13.76 

5.98 

13.73  t 

6.04 

15 

16 

14.73 

6.25 

14.70 

6.32 

14.67 

6.38 

14.64 

6.44 

16 

17 

15.65 

6.64 

15.62 

6.71 

15.59 

6.78 

15.56 

6.85 

17 

18 

16.57 

7.03 

16.54 

7.11 

16.51 

7.18 

16.48 

7.25 

18 

19 

17.49 

7.42 

17.46 

7.50 

17.42 

7.58 

17.39 

7.65 

19 

20 

18.41 

7.81 

18.38 

7.89 

18.34 

7.97 

18.31 

8.05 

20 

21 

19.33 

8.21 

19.29 

8.29 

19.26 

8.37 

19.22 

8.46 

21 

22 

20.25 

8.60 

20.21 

8.68 

20.18 

8.77 

20.14 

8.86 

22 

23 

21. 17  | 

8.99 

I 21.13 

9.08 

21.09 

9.17 

21.05 

9.26 

23 

24 

22.09  | 

9.38 

22.05 

9.47  | 

I 22.01 

9.57 

21.97 

9.67 

24 

25 

23.01 

9.77 

22.97 

9.87  1 

22.93 

9.97 

22.88 

10.07 

-25 

26 

23.93 

*0.16 

23.89 

10.26  j 

23.84 

10.37 

23.80 

10.47 

26 

27 

24.85 

.0.55 

24.81 

10.66 

24.76 

10.77 

24.71 

10.87 

27 

28 

25.77 

.0.94 

25.73 

11.05 

25.68 

11.16 

25.63 

11.28 

28 

29 

26.69 

11.33 

26.64 

11.45 

26.59 

11.56 

26.54 

11.68 

29 

30 

27.62 

11.72 

27.56 

11.84 

27.51 

11.96 

27.46 

12.08 

30 

31 

28.54 

12.11 

28.48 

12.24 

28.43 

12.36 

28.37 

12.49 

31 

32 

29.46 

12.50 

29.40 

12.63 

29.35 

12.76 

29.29 

12.89 

32 

33 

30.38 

12.89 

30.32 

13.03 

30.26 

13.16 

30.21 

13.29 

33 

34 

31.30 

13.28 

31.24 

13.42 

31.18 

13.56 

31.12 

13.69 

34 

35 

32.22 

13.68 

32.16 

13.82 

32.10 

13.96 

32.04 

14.10 

35 

36 

33.14 

14.07 

33.08 

14.21 

33.01 

14.35 

32.95 

14.50 

36 

37 

34.06 

14.46 

34.00 

14.61 

33.93 

14.75 

33.87 

14.90 

37 

38 

, 34.98 

14.85 

34.91 

15.00 

34.85 

15.15 

34.78 

15.30 

38 

39 

35.90 

15.24 

35.83 

15.39 

35.77 

15.55 

35.70 

15.71 

1 39 

40 

36.82 

15.63 

36.75 

15.79 

36.68 

15.95 

36.61 

16.11 

40 

41 

37.74 

16.02 

37.67 

16.18 

37.60 

16.35 

37.53 

16.51 

11 

42 

38.66 

16.41 

38.59 

16.58 

38.52 

16.75 

38.44 

16.92 

42 

43 

39.58 

| 16.80 

39.51 

16.97 

39.43 

17.16 

39.36 

17.32 

43 

' 44 

j 40  50 

i 17. 19 

40.43 

17.37 

40.35 

17.54 

40.27 

17.72 

44 

1 c 

41  42 

! 17.58 

41.35 

17.76 

41.27 

17.94 

41.19 

18.12 

45 

46 

42.34 

1 17.97 

42.26 

18.16 

42.18 

18.34 

42.10 

18.53 

46 

47 

1 43.26 

18.36 

43.18 

18.55 

43.10 

18.74 

43.02 

18.93 

47 

48 

41.18 

j 18.76 

44.10 

18.95 

44.02 

19.14 

43.93 

19.33 

48 

19 

45.10 

! 19.15 

45.02 

19.34 

44.94 

19.54 

44.85 

19.73 

49 

50 

4"  .03 

, 19.54 

45.94 

| 19.74 

45.85 

19.94 

45.77 

20.14 

50 

f 

Dep. 

I Lat. 

Dop. 

Lat. 

Dop. 

Lat. 

I Dep. 

Lat. 

6 

o 

e 

o 

07 

Dog 

| 60f  Dog. 

66*  Dog. 

66*  Dog. 

od 

Q 

TRAVER8E  TABLE 


119 


3 

S' 

23  Dog 

23}  Deg. 

23} 

Deg. 

23}  Deg. 

So" 

03 

3 

o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

o 

? 

51 

46.95 

19.93 

46.86 

20.  13 

46.77 

20.34 

46.68 

20.54 

; 5i 

52 

47  87 

20.32 

47.78 

20 . 53 

47.69 

20.73 

47.60 

20.94 

i 52 

53 

48.79 

20.71 

48.70 

20.92 

48.60 

21.13 

48.51 

21.35 

i 53 

54 

49.71 

21.10 

49.61 

21.32 

49.52 

21.53 

49  43 

21.75 

54 

55 

50.63 

21.49 

50.53 

21.71 

50.44 

21.93 

50.34  ! 22.15 

55 

56 

51.55 

21.88 

51.45 

22.11 

51.36 

22.33 

51.26 

22.55 

1 56 

57 

52.47 

22.27 

52.37 

22.50 

52.27 

22.73 

52.17 

22.96 

57 

58 

53.39 

22.66 

53.29 

22.90 

53.19 

23.13 

53.09 

23.36 

58 

59 

54.31 

23.05 

54.21 

23.29 

54.11 

23.53 

54  00 

23.76 

59 

80 

55.23 

23.44 

55.13 

23.68 

55.02 

23.92 

54.92 

24.16 

60 

61 

56.15 

23.83 

56.05 

24.08 

55.91 

24.32 

55.83 

24.57 

61 

62 

57.07 

24.23 

56.97 

24  47 

56.86 

24.72 

56.75 

24.97 

62 

63 

57.99 

24.62 

57.88 

24.87 

57.77 

25.12 

57.66 

25.37 

63 

64 

58.91 

25.01 

58.80 

25.26 

58.69 

25.52 

58.58 

25 . 78 

64 

65 

59.83 

25.40 

59.72 

25.66 

59.61 

25.92 

59.50 

26.18 

65 

66 

60.75 

25.79 

60.64 

26.05 

60.53 

26.32 

60.41 

26.58 

66 

67 

61.67 

26.18 

61.56 

26-45 

61.44 

26.72 

61.33 

26.98 

67 

68 

62.59 

26.57 

62.48 

26.84 

62.36 

27.11 

62.24 

27.39 

68 

69 

63.51 

26.96 

63.40 

27.24 

63.28 

27.51 

63.16 

27.79 

69 

70 

64.44 

27.35 

64.32 

27.63 

64.19 

27.91 

64.07 

28.19 

70 

71 

65.36 

27.74 

65.23 

28.03 

65.11 

28.34 

64.99 

28.59 

71 

72 

66.28 

28.13 

66.15 

28.42 

66.03 

28.71 

65.90 

29.00 

72 

73 

67.20 

28.52 

67.07 

28.82 

66.95 

29.11 

66.82 

29.40 

73 

74 

68.12 

28.91 

67.99 

29.21 

67.86 

29.51 

67.73 

29.80 

74 

75 

69.04 

29.30 

68.91 

29.61 

68.78 

29.91  | 

68.65 

30.21 

75 

76 

69.96 

29.70 

69.83 

30.00 

69.70 

30.30 

69.56 

30.61 

76 

77 

70.88 

30.09 

70 . 75 

30.40 

70.61 

30.70 

70.48 

31.01 

77 

78 

71.80 

30.48 

ri.67 

30.79 

71.53 

31.10 

71.39 

31.41 

78 

79 

72.72 

30.87 

72.58 

31.18 

72.45 

31.50 

72.31 

31.82 

79 

80 

73.64 

31.26 

73.50 

31.58 

73.36 

31.90 

73.22 

32.22 

80 

81 

74.56 

31.65 

74.42 

31.97 

74.28 

32.30 

74.14 

32.62 

'81 

82 

75.48 

32.04 

75.34 

32.37 

75.20 

32.70 

75.06 

33.03 

82 

83 

76.40 

32.43 

76.26 

32.76 

76.12 

33.10 

75.97 

33.43 

83 

84 

77.32 

32.82 

77.18 

33.16 

77.03 

33.49 

76.89 

33.83 

84 

85 

78.24 

33.21 

78.10 

33.55 

77.95 

33.89 

77.80 

34.23 

85 

86 

79.16 

33 . 60 

79.02 

33.95 

78.87 

34.29 

78.72 

34.64 

86 

87 

80.08 

33.99 

79.93 

34.34 

79.78 

34.69 

79.63 

35.04 

87 

88 

81.00 

34.38 

80.85 

34.74 

80.70 

35.09 

80.55 

35.44 

88 

89 

81.92 

34.78 

81.77 

35.13 

81.62 

35.49 

81.46 

35.84 

89 

90 

82.85 

35.17 

82.69 

35.53 

82.54 

35.89 

82.38 

36.25  1 

90 

91 

83.77 

35.56 

83.61 

35 . 92 

83.45  i 

36.29 

83.29 

36.65  ! 

91 

92 

84.69 

35 . 95 

84.53 

36.32 

84.37  , 

36.68 

84.21 

37.05 

92 

93 

85.61  | 

36.34 

85.45 

36.71 

85.29 

37.08 

85.12 

37.46 

93 

94 

86.53 

36.73 

86.37 

37.11 

86.20 

37.48 

86.04 

37.86  1 

94 

95 

87.45 

37. 12 

87.29 

37.50 

87.12 

37.88 

86 . 95 

38.26 

95 

96 

88.37 

37.51 

88.20 

37.90 

88.04 

38.28 

87.87 

38.66 

96 

97 

89.29 

37.90 

89. 12 

38.29 

88.95 

38.68 

88.79 

39.07 

97 

98 

90.21 

38.29 

90.04 

38 . 68 

89.87 

39.08 

89.70 

39.47 

98 

99 

91.13 

38.68 

90.96 

39.08 

90.79 

39.48 

90.62 

39.87 

99 

100 

92.05 

39.07 

91.88 

39.47 

91.71 

39.87 

91.53 

40.27 

100 

\\ 

Dep.  ; 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep 

Lat. 

8 

a 

5 I 

67  Deg. 

66}  Deg. 

661 

Deg. 

66* 

2 

s 

a / 

i 

L20 


TRAVRRAE  TABLE, 


Distance.! 

24  Dog. 

24*  Deg. 

LIA\  Dog. 

24|  Deg. 

1"J 

ll 

Lat. 

j Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

8 

1 

0.91 

1 0.41 

0.91 

0.41 

0.91 

0.41 

0,91 

0.42 

l 

2 

1.83 

0.81 

1.82 

0.82 

1.82 

0.83 

1.82 

0.84 

2 

3 

2.74 

1 .22 

2.74 

1.23 

2.73 

1.24 

2 72 

l .26 

3 

4 

3.65 

1.63 

3.65 

1.64 

3.64 

1.66 

3.63 

1.67 

4 

5 

4.67 

2.03 

4.56 

2.05 

4.55 

2.07 

4.64 

2.09 

6 

6 

5.48 

2.44 

5.47 

2.46 

5.46 

2.49 

5.45 

2.51 

6 

7 

i 6.39 

2.85 

6.38 

2.87 

6.37 

2.90 

6.36 

2.93 

7 

8 

7.31 

3.25 

7.29 

3.29 

7.28 

3.32 

7.27 

3.35 

8 

9 

8.22 

! 3.66 

8.21 

3.70 

8.19 

3.73 

8.17 

3.77 

9 

10 

9.14 

4.07 

9.12 

4.11 

9.10 

4.15 

9.08 

4.19 

rO 

11 

10.05 

4.47 

10.03 

4.52 

10.01 

4.56 

9.99 

4.61 

11 

12 

10.96 

4.88 

10.94 

4.93 

10.92 

4.98 

10.90 

5.02 

12 

13 

11.88 

5.29 

11.85 

5.34 

11.83 

5.39 

11.81 

5.44 

13 

14 

12.79 

5.69 

12.76 

6.75 

12.74 

5.81 

12.71 

6.86 

14 

15 

13.70 

6.10 

13.68 

6.16 

13.65 

6.22 

13.62 

6.28 

15 

16 

14.62 

6.51 

14.59 

6.57 

14.56 

6.64 

14.63 

6.70 

16 

17 

15.53 

6.92 

15.50 

6.98 

15.47 

7.05 

16.44 

7.12 

17 

18 

16.44 

7.32 

16.41 

7.39 

16.38 

7.46 

16.35 

7.64 

18 

19 

17.36 

7.73 

17.32 

7.80 

17.29 

7.88 

17.26 

7.95 

19 

20 

18.27 

8.13 

18.24 

8.21 

18.20 

8.29 

18.16 

8.37 

20 

21 

19.18 

8.54 

19.15 

8.63 

19.11 

8.71 

19.07 

8.79 

21 

22 

20.10 

8.95 

20.06 

9.04 

20.02 

9.12 

19.98 

9.21 

22 

23 

21.01 

9.35 

20.97 

9.45 

20.93 

9.54 

20.89 

9.63 

23 

24 

21.93 

9.76 

21.88 

9.86 

21.84 

9.95 

21.80 

10.05 

24 

25 

22.84 

10.17 

22.79 

10.27 

22.75 

10.37 

22.70 

10.47 

25 

26 

23.75 

10.58 

23.71 

10.68 

23.66 

10.78 

23.61 

10.89 

26 

27 

24.67 

10.98 

24.62 

11.09 

24.57 

11.20 

24.52 

11.30 

27 

28 

25.58 

11.39 

25.53 

11.50 

25.48 

11.61 

25.43 

11.72 

28 

29 

26.49 

11.80 

26.44 

11.91 

26.39 

12.03 

26.34 

12.14 

29 

30 

27.41 

12.20 

27.35 

12.32 

27.30 

12.44 

27.24 

12.56 

30 

31 

28.32 

12.61 

28.26 

12.73 

28.21 

12.86 

28.15 

12.98 

31 

32 

29.23 

13.02 

29.18 

13.14 

29.12 

13.27 

29.06 

13.40 

32 

33 

30.15 

13.42 

30.09 

13.55 

30.03 

13.68 

29.97 

13.82 

33 

34 

31.06 

13.83 

31.00 

13.96 

30.94 

14.10 

30.88 

14.23 

34 

35 

31.97 

14.24 

31.91 

14.38 

31.85 

14.51 

31.78 

14.65 

35 

36 

32.89 

14.64 

32.82 

14.79 

32.76 

14.93 

32.69 

15.07 

36 

37 

33.80 

15.05 

33.74 

15.20 

33.67 

15.34 

33.60 

15.49 

37 

38 

34.71 

15.46 

34.65 

15.61 

34.58 

15.76 

34.51 

15.91 

38 

39 

35.63 

15.86 

35.56 

16.02 

35.49 

16.17 

35.42 

16.33 

39 

40 

36.54 

16.27 

36.47 

16.43 

36.40 

16.59 

36.33 

16,75 

40 

'41 

37.46 

16.68 

37.38 

16.84 

37.31 

17.00 

37.23 

17.16 

il 

42 

38.37 

17.08 

38.29 

17.25 

38.22 

17.42 

38.14 

17.58 

12 

43 

39.28  1 

17.49 

39.21 

17.66 

39.13 

17.83 

39.05 

18.00 

13 

44 

40.20 

17.90  : 

40.12 

18.07 

40.04 

18.25 

39.96 

18.42 

44 

45 

41.11  j 

18  30 

41.03 

18.48 

40.95 

18.66 

40.87 

18.84 

45 

46 

42.02  r 

18.71 

41.94 

18.89 

41.86 

19.08 

41.77 

19.26 

46 

47 

42.94  19.12 

42.85 

19.30 

42.77 

19.49 

42.68 

19,68 

47 

48 

43.86  ' 

19.52 

43.76 

19.71 

43.68 

19.91 

43.59 

20.10 

48 

49 

44.76 

19.93 

44.68 

20.  13 

44.59 

20.32 

44.50 

20.51 

19 

50 

45.68 

20.34 

4ft.  59 

20.54 

45.50 

20.73 

_45.41_ 

20.93 

60 

8 

5 

Dop.  | 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

« 

o 

c 

5 

QD 

6 

66  Dog  | 

65.?  Deg. 

65 H Dog. 

654  Deg 

5 

cn 

5 

. . 

TRAVERSE  TARI.E. 


121 


p 

55 

24  Deg. 

24$  Deg. 

24£  Deg. 

24|  Deb 

P 

on 

P 

3 

a 

p 

Licit* 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

a 

51 

46.59 

20.74 

46.50 

20.95 

46.41 

21.15 

46732 

irr.35 

‘si 

52 

47.50 

21.15 

47.41 

21.36 

47.32 

21.56 

47.22 

21.77 

52 

53 

48.42 

21.56 

48.32 

21.77 

48.23 

21.98 

48.13 

22.19 

53 

54 

49.33 

21.96 

49.24 

22.18 

49.14 

| 22.39 

49.04 

22.61 

54 

55 

50.24 

22.37 

50.15 

22.59 

50.05 

22.81 

49.95 

23.03 

56 

56 

51.16 

22.78 

51.06 

23.00 

50.96 

1 23.22 

50.86 

23.44 

56 

57 

52.07 

23.18 

51.97 

23.41 

51.87  1 23.64 

51.76 

23.86 

57 

58 

52.99 

23.59 

52.88 

23.82 

52.78 

24.05 

52.67 

24.28 

58 

59 

53.90 

24.00 

53.79 

24.23 

53.69 

24.47 

53.58 

24.70 

59 

60 

54.81 

24.40 

54.71 

24.64 

54.60 

24.88 

54.49 

25.12 

60 

61 

55.73 

24.81 

55.62 

25.05 

55.51 

25.30 

55.40 

25.54 

61 

62 

56.64 

25.22 

56.53 

25.46 

56.42 

25.71 

56.30 

25.96 

62 

63 

57.55 

25.62 

57.44 

25.88 

57.33 

26.13 

57.21 

26.38 

63 

64 

58.47 

26.03 

58.35 

26.29 

58.24 

26.54 

58.12 

26.79 

64 

65 

59.38 

26.44 

59.26 

26.70 

59.15 

26.96 

59.03 

27.21 

65 

66 

60.29 

26.84 

60.18 

27.11 

60.06 

27.37 

59.94 

27.63 

66 

67 

61.21 

27.25 

61.09 

27.52 

60.97 

27.78 

60.85 

28.05 

67 

68 

62.12 

27.66 

62.00 

27.93 

61.88 

28.20 

61.75 

28.47 

68 

69 

63.03 

28.06 

62.91 

28.34 

62.79 

28.61 

62.66 

28.89 

69 

70 

63.95 

28.47 

63.8-2 

28.75 

63.70 

29.03 

63.57 

29.31 

70 

71 

64.86 

28.88 

64.74 

29.16 

64.61 

29.44 

64.48 

29.72 

71 

72 

65.78 

29.28 

65.65 

29.57 

65.52 

29.86 

65.39 

30.14 

72 

73 

66.69 

29.69 

66.56 

29.98 

66.43 

30.27 

66.29 

30.56 

73 

74 

67.60 

30.10 

67.47 

30.39 

67.34 

30.69 

67.20 

30.98 

74 

75 

68.52 

30.51 

68.38 

30.80 

68.25 

31.10 

68.11 

31.40 

75 

76 

69.43 

30.91 

69.29 

31.21 

69.16 

31.52 

69.02 

31.82 

76 

77 

70.34 

31.32 

70.21 

31.63 

70*.  07 

31.93 

69.93 

32.24 

77 

78 

71.26 

31.73 

71.12 

32.04 

70.98 

32.35 

70.84 

32.66 

78 

79 

72.17 

32.13 

72.03 

32.45 

71.89 

32.76 

71.74 

33.07 

79 

80 

73.08 

32.54 

72.94 

32.86 

72.80 

33.15 

72.65 

33.49 

80 

81  i 

74.00 

32.95 

73.85 

33.27 

73.71 

33.59 

73.56 

33.91 

81 

82 

74.91 

33.35 

74.76 

33.68 

74.62 

34.00 

74.47 

34.33 

82 

83 

75.82 

33.76 

75.68 

34.09 

75.53 

34.42 

75.38 

34.75 

83 

84 

76.74 

34.17 

76.59 

34.50 

76.44 

34.83 

76.28 

35.17 

84 

85 

77.65 

34.57 

77.50 

34.91 

77.35 

35.25 

77.19 

35.59 

85 

86 

78.56 

34.98 

78.41 

35.32 

78  26 

35.66 

78.10 

36.00 

86 

87 

79.48 

35.39 

79.32 

35.73 

79.17 

36.08 

79.01 

36.42 

87 

88 

80.39 

35.79 

80.24 

36.14 

80.08 

36.49 

79.92 

36.84 

88 

89 

81.31 

36.20 

81.15 

36.55 

80.99 

36.91 

80.82 

37  26 

89 

90 

82.22 

36.61 

82.06 

36.96 

81.90 

37.32 

81.73 

37.68 

90 

9 L 

83.13 

37.01 

82.97 

37.38 

82.81 

37.74 

82.64 

38.10 

91 

92 

84.05 

37.42 

83.88 

37.79 

83.72 

38.15 

83.55 

38.52 

92 

93 

84  96 

37.83 

84.79 

38.20 

84.63 

38.57 

84.46 

38.94 

93 

94 

85  87 

38.2? 

85.71 

38.61 

85.54 

38.98 

85.37 

39.35 

94 

, 95 

86  79 

38.64 

86.62 

39.02 

86.45 

39.40 

86.27 

39  77 

95 

96  87  70 

39.05  ; 

87.53 

39.43 

87.36 

39.81 

87.18 

40.19 

96 

97 

88.61 

39.  45 

88.44 

39.84 

88.27 

40.23 

88.09 

40.61 

97 

08  , 

89.53 

39  06 

89.35 

40.25 

89.18 

40.64 

89.00 

41.03 

98 

99 

90.44 

40.27  ; 

90.26 

40.66 

90.09 

41.05 

89.91 

41.45 

99 

00 

91 . 35 

40.67 

91.18 

41.07 

91.00 

41.47 

90.81 

41.87 

100 

l 

Dep. 

Let. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

c! 

i 

66  Deg. 

65$  Deg. 

65J  Deg. 

65*  Dog. 

5 

122 


TRAVERSE  TAJILE. 


o 

5 

p 

25  Deg. 

25$  Dog. 

26  b Deg 

26$  Deg. 

f 

3 

o 

CD 

L&t. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.91 

0.42 

9.90 

0.43 

0790 

0.43 

0.90 

0.43 

2 

1.81 

0.85 

1.81 

0.85 

1.81 

0.86 

1.80 

0.87 

8 

3 

2.72 

1.27 

2.71 

1.28 

2.71 

1.29 

2.70 

1.30 

3 

4 

3.63 

1.69 

3.62 

1.71 

3.61 

1.72 

3.60 

1.74 

4 

5 

4.53 

2.11 

4.52 

2.13 

4.61 

2.15 

4.50 

2.17 

5 

6 

5.44 

2.54 

5.43 

2.56 

5.42 

2.58 

5.40 

2.61 

6 

7 

6.34 

2.96 

6.33 

2.99 

6.32 

3.01 

6.30 

3.04 

7 

8 

7.25 

3.38 

7.24 

3.41 

7.22 

3.44 

7.21 

3.48 

8 

9 

8.16 

3.80 

8.14 

3.84 

8.12 

3.87 

8.11 

3.91 

9 

10 

9.06 

4.23 

9.04 

4.27 

9.03 

4.31 

9.01 

4.34 

10 

11 

9.97 

4.65 

9.95 

4.69 

9.93 

4.74 

“ 9.91 

4.78 

11 

12 

10.88 

5.07 

10.85 

5.12 

10.83 

5.17 

10.81 

5.21 

12 

13 

11.78 

5.49 

11.76 

5.55 

11.73 

5.60 

11.71 

5.65 

13 

14 

12.69 

5.92 

12.66 

5.97 

12.64 

6.03 

12.61 

6.08 

14 

15 

13.59 

6.34 

13.57 

6.40 

13.54 

6.46 

13.51 

6.52 

15 

16 

14.50 

6.76 

14.47 

6.83 

14.44 

6.89 

14.41 

6.95 

16 

17 

15.41 

7.18 

15.38 

7.25 

15.34 

7.32 

15.31 

7.39 

17 

18 

16.31 

7.61 

16.28 

7.68 

16.25 

7.75 

16.21 

7.82 

18 

19 

17.22 

8.03 

17. 18 

8.10 

17.15 

8.18 

17.11 

8.25 

19 

20 

18.13 

8.45 

18.09 

8.53 

18.05 

8.61 

18.01 

8.69 

20 

21 

19.03 

8.87 

18.99 

8.96 

18.95 

9.04 

18.91 

9.12 

21 

22 

19.94 

9.30 

19.90 

9.38 

19.86 

9.47 

19.82 

9.56 

22 

23 

20.85 

9.72 

20.80 

9.81 

20.76 

9.90 

20.72 

9.99 

23 

24 

21 .75 

10.14 

21.71 

10.24 

21.66 

10.33 

21.62  j 

10.13 

24 

25 

22.66 

10.57 

22.61 

10.66 

22.56 

10.76 

22.52  j 

10.86 

25 

26 

23.56 

10.99 

23.52 

11.09 

23.47 

11.19 

23.42 

11.30 

26 

27 

24.47 

11.41 

24.42 

11.52 

24.37 

11.62 

24.32 

11.73 

27 

28 

25.38 

11.83 

25.32 

11.94 

25.27 

12.05 

25.22 

12.16 

28 

29 

26.28 

12.26 

26.23 

12.37 

26.17 

12.48 

26.12 

12.60 

29 

30 

27.19 

12.68 

27.13 

12.80 

27.08 

12.92 

27.02 

13.03 

30 

31 

28.10 

13. 10 

28.04 

13.22 

27.98 

13.35 

27.92 

13.47 

31 

32 

29.00 

13.52 

28.94 

13.65 

28.88 

13.78 

28.82 

13.90 

32 

33 

29.91 

13.95 

29.85 

14.08 

29.79 

14.21 

29.72 

14.34 

33 

34 

30.81 

14.37 

30.75 

14.50 

30.69 

14.64 

30.62 

14.77 

34 

35 

31.72 

14.79 

31.66 

14.93 

31.59 

15.07 

31.52 

15.21 

35 

36 

32.63 

15.21 

32.56 

15.36 

32.49 

15.50 

32.43 

15.64 

36 

37 

33.53 

15.64 

33.46 

15.78 

33.40 

15.93 

33.33 

16.07 

37 

38 

34.44 

16.06 

34.37 

16.21 

34.30 

16.36 

34.23 

16.51 

36 

39 

35.35 

16.48 

35.27 

16.64 

35.20 

16.79 

35.13 

16  94 

39 

40 

36.25 

16.90 

36.18 

17.06 

36.10 

17.22 

36.03 

17.38 

40 

41 

37.16 

17.33 

37.08 

17.49 

37.  Of 

17.65 

36.93 

17.81 

41 

42 

38.06 

17.75 

37.99 

17.92 

37.91 

18.08 

37.83 

18.25 

42 

43 

38.97 

18.17 

38.89 

18.34 

38.81 

18.51 

38.73 

18.68 

43 

44 

39.88 

18.60 

39.80 

18.77 

39.71 

18.94 

39.63 

19.12 

44 

45 

40.78 

19.02 

40.70 

19.20 

40.62 

19.37 

40.53 

19.65 

45 

46 

41.69 

19.44 

41.60 

19.62 

41.52 

19.80 

41.43 

19.98 

46 

47 

42 , 60 

19.86 

42.51 

20.05 

42.42 

20.23 

42.33 

20.42 

1 47 

48 

43.50 

20.29 

43.41 

20.48 

43.32 

20.66 

43.23 

20.85 

48 

49 

44.41 

20.71 

44.32 

20.90 

44.23 

21.10 

44.13 

21.29 

1 49 

50 

45.32 

21.13 

45.22 

21.33 

45.13 

21.53 

46.03 

21.72 

60 

Distance. j 

Dop. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

V 

c 

a 

65  Deg. 

64|  Dog. 

645  Dog. 

64$  Dog. 

s 

.a 

Q 

TRAVERSE  TABLE. 


123 


o 

S* 

25  Deg. 

25!  Deg 

25$  Deg. 

25|  Dog. 

, c 

1 “ 

£ 

s 

© 

© 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

o 

* 

51 

4tL22 

21.55 

46.13 

21.75 

46.03 

21.96 

45.94 

22.16 

i 51 

52 

47.13 

21.98 

47.03 

22.18 

46.93 

22.39 

46.84 

22.59  ; 

! 52 

53 

48.03 

22.40 

47.94 

22.61 

47.84 

22.82 

47.74 

23.03 

53 

54 

48.94 

22.82 

48.84 

23.03 

48.71 

23.25 

48.64 

23.46  1 

54 

55 

49.85 

23.24 

49.74 

23.46 

49.64 

23.68 

49.54 

23.89  ! 

t 55 

56 

50.75 

23.67 

50 . 65 

23.89 

50.54 

24.11 

50.44 

24.33 

56 

51 

51  66 

24.09 

51.55 

24.31 

51.45 

24.54 

51.34 

24.76 

57 

58 

52.57 

24.51 

52.46 

24.74 

52.35 

24.97 

52.24 

25.20 

58 

59 

53.47 

24.93 

53,36 

25.17 

53.25 

25.40 

53.14 

25.63 

59 

60 

54.38 

25.36 

54.27 

25.59 

54.16 

25.83 

54.04 

26.07 

60 

61 

55.28 

25.78 

55.17 

26.02 

55.06 

26 . 26 

54.94 

26.50 

'61 

62 

56.19 

26.20 

56.08 

26.45 

55.96 

26.69 

55.84 

26.94 

62 

63 

57.10 

26  62 

56 . 98 

26.87 

56.86 

27.12 

36 . 74 

27  37 

63 

64 

58.00 

27.05 

57.89 

27.30 

57.77 

27.55 

57.64 

27.80 

64 

65 

58.91 

27.47 

58.79 

27.73 

58.67 

27.98 

58.55 

28.24 

65 

66 

59.82 

27.89 

59 . 69 

28.15 

59.57 

28.41 

59.45 

28.67 

66 

67 

60.72 

28.32 

60.60 

28.58 

60.47 

28.84 

60.35 

29.11 

67 

68 

61.63 

28.74 

61.50 

29.01 

61.38 

29.27 

61.25 

29.54 

68 

69 

62.54 

29.16 

62.41 

29.43 

62.28 

29.71 

62.15 

29.98 

69 

70 

63.44 

29.58 

63.31 

29.86 

63.18 

30.14 

63.05 

30.41 

70 

71 

64.35 

30.01 

64.22 

30.29 

64.08 

30.57 

63.95 

30.85 

71 

72 

65.25 

30.43 

65.12 

30.71 

64.99 

31.00 

64.85 

31 .28 

72 

73 

66.16 

30.85 

66.03 

31.14 

65.89 

31.43 

65.75 

31.71 

73 

74 

67.07 

31.27  1 

66.93 

31.57 

66.79 

31 .86 

66.65 

32.15  j 

74 

75 

67.97 

31.70 

67.83 

31.99 

67.69 

32.29 

67.55 

32.58 

75 

76 

68.88 

32.12 

68.74 

32.42 

68.60 

32.72 

68.45 

33.02  1 

76 

77 

69.79 

32.54 

69.64 

32.85 

69.50 

33.15 

69.35 

33.45  | 

77 

78 

70.69 

32.96 

70.55 

33.27 

70.40 

33.58 

70.25 

33.89  i 

78 

79 

71.60 

33.39 

71.45 

33.70 

71 .30 

34.01 

71.16 

34.32  1 

79 

80 

72.50 

33.81 

72.36 

34.13 

72.21 

34.44 

72.06 

34.76  | 

80 

81 

73.41 

34.23 

73.26 

34.55 

73.11 

34.8/ 

72.96 

35. 19 

81 

82 

74.32 

34.65 

74.17 

34.98 

74.01 

35.30 

73.86 

35.62 

82 

83 

75.22 

35.08 

75.07 

35.41 

74.91 

35.73 

74.76 

36.06 

83 

84 

76.13 

35.50 

75.97 

35.83 

75.82 

36.16 

75.66 

36.49 

84 

85 

77.04 

35.92 

76.88 

36.26 

76.72 

36.58 

76.56 

36.93 

85 

86 

77.94 

36.35 

77.78 

36.68 

77.62 

37.02 

77.46 

37.36 

86 

87 

78.85 

36.77 

78.69 

37.11 

78.52 

37.45 

78.36 

37.80 

87 

88 

79.76 

37.19 

79.59 

37.54 

79.43 

37.88 

79.26 

I 38.23 

88 

89 

80.66 

37.61 

80.50 

37.96 

80.33 

38.32 

80.16 i 

i 38.67 

89 

90 

81.57 

38.04 

81.40 

38.39 

81.23 

38.75 

81.06 

39.10 

. 90 

91 

82.47 

38 .46 

82.31 

38.82 

82.14 

39.18 

81.96 

39.53 

.91 

92 

83.38 

38.88 

83.21 

39.24 

83.04 

39.61 

82.86 

39.97 

| 92 

93 

84.29 

39.30 

84.11 

39.67 

83.94 

40.04 

83.76 

40.40 

! 93 

94 

85.19 

39.73 

85.02 

40.10 

84.84 

40.47 

84.67 

40.84 

94 

95 

86.10 

40.15 

85.92 

40.52 

! 85.75 

40.90 

85  57 

41.27 

95 

96 

87.01 

40.57 

86.83 

40.95 

; 86.65 

41.33 

86  47 

41.71 

96 

97 

87.91 

40.99 

87.73 

41.38 

87.55 

| 41.76 

87.37 

I 42.14 

97 

9b 

88.82 

41.42  ■ 

88.64 

41.80  1 

88.45 

42.19 

88.27 

42.58 

98 

99 

; 89.72  141.84 

89.54 

42.23  ! 

89.36 

1 42.62 

89.17 

1 43.01 

99 

100 

99.63 

1 42.26  | 

90.45 

42.66 

90.26 

| 43.05 

90.07 

,43  44 

A00 

d 

y 

c 

Dep. 

I Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| lat. 

d 

o 

c 

03 

® 

3 

65  Deg. 

64f  Deg. 

64!  Deg. 

64!  Deg. 

rf 

a 

cl 

» 

124 


l'K  AV  EK££  'lAHLt 


1 

g ’ 

5 , 
S 1 

20  Dog. 

264  Deg. 

26  £ Dog.  | 

26?  Deg. 

O 

S’ 

7 

D 

£ 

I.  at. 

Dep. 

Lat. 

Dop. 

Lat.  | 

Dep. 

Lat. 

Dop. 

5 

3 

1 

0.90 

044 

0.90  1 

0.44 

0.89! 

0.45 

oM 

~0 .45 

1 

2 

1.80 

9.88 

1.79  ! 

0.88 

1.79  1 

0.89 

1.79 

0.90  i 

2 

3 ! 

2 70 

1.32 

2 69 

1.33 

2.68  j 

1.34 

2.68 

1.35 

3 

4 

3.60 

l 75 

3.59 

1.77 

3.58  I 

1.78 

3.57 

1.80 

1 

l 

4.49 

2 19 

4.48 

2.21 

4.47 

2.23 

4.46 

2.25 

5 

6 

5.39 

2.63 

5.38 

2.65 

5.37 

2.68 

5.36 

2.70 

6 

7 

6.29 

3.07 

6.28 

3.10 

6.26 

3.12 

6.25 

3.15 

7 

8 

7.19 

3.51 

7.17 

3.54 

7.  16 

3.57 

7.14 

3.60 

8 

9 

8.09 

3.95 

8.07 

3.98 

8.05 

4.02 

8.04 

4.05  , 

9 

10 

8.99 

4.38 

8.97 

4.42 

8.95 

4.46 

8.93 

4.50 

10 

LI 

9.89 

4.82 

9.87 

4.87 

9.84 

4.91 

9.82 

4.95 

11 

12 

10.79 

5.26 

10.76 

5.31 

10.74 

5.35 

10.72 

5.40 

12 

13 

11.68 

5.70 

11.66 

5.75 

11.63 

5.80 

11.61 

5.85 

13 

14 

12.58 

6.14 

12.56 

6. 19 

12.53 

6.25 

12.50 

6.30 

14 

15 

13.48 

6.58 

13.45 

C 63 

13.42 

6.69 

13.39 

6.75 

15 

16 

14.38 

7.01 

14.35 

7.08 

14.32 

7.14 

14.29 

7.20 

16 

17 

15.28 

7.45 

15.25 

7.52 

15.21 

7.59 

15.18 

7.65 

17 

18 

16.18 

7.89 

16.14 

7.96 

16.11 

8.03 

16.07 

8.10 

18 

19 

17.08 

8.33 

17.04 

8.40 

17.00 

8.48 

16.97 

8.55 

19 

20 

17.98 

8.77 

17.94 

8.85 

17.90 

8.92  1 

17.86 

9.00 

20 

21 

18.87 

9.21 

18.83 

9.29 

18.79 

9.37  | 

18.75 

9.45 

21 

22 

19.77 

9.64 

19.73 

9.73 

19.09 

9.82  1 

19.65 

9.90 

22 

23 

20.67 

10.08 

20.63 

10.17 

20.58 

10.26 

20.54 

10.35 

23 

24 

21.57 

10.52 

21.52 

10.61 

21.48 

10.71 

21.43 

10.80 

24 

25 

22.47 

10.96 

22.42 

11.06 

22.37 

11.15 

22.32 

11.25 

25 

26 

23.37 

11.40 

23.32 

11.50 

23.27 

11.60 

23.22 

11.70 

26 

27 

24.27 

11.84 

24.22 

11.94 

24.16 

12.05 

24.11 

12.15 

27 

28 

25. 17 

12.27 

25.11 

12.38 

25.06 

12.49 

25 . 00 

12.60 

28 

29 

26.06 

12.71 

26.01 

12.83 

25.95 

12.94 

25.90 

13.05 

29 

30 

26.90 

13.15 

26.91 

13.27 

26.85 

13.39 

26.79 

13.50 

I 30 

SI 

27.86 

13.59 

27.80 

13.71 

27.74 

13.83 

27.68 

13.95 

31 

32 

28.76 

14.03 

28.70 

14.15 

28.64 

14.28 

28.58 

j 14.40 

32 

33 

29.66 

i 14.47 

29.60 

14.60 

29.53 

14.72 

29.47 

14.85 

33 

34 

i 30.56 

14.90 

30.49 

15.04 

30.43 

15.17 

30.36 

| 15.30 

34 

35 

131.46 

15.34 

31.39 

15.48 

31.32 

15.62 

31.25 

15.75 

35 

36 

32.36 

! 15.78 

32.29 

15.92 

32.22 

16.06 

32.15 

16.20 

36 

37 

1 33^6 

16.22 

33.18 

16.36 

33.11 

16.51 

33.04 

16.65 

37 

38 

34.15 

16.66 

34.08 

16.81 

34.01 

16.96 

33.93 

17.10 

38 

39 

35.05 

17.10 

34.98 

17.25 

34.90 

17.40 

34.83 

17.55 

39 

40 

35.95 

j 17.53 

35.87 

17.69 

35.80 

17.85 

35.72 

18.00 

40 

41 

36.85 

1 17.97 

36.77 

18.13 

36.69 

18.29 

36.61 

18.45 

41 

42 

37.75 

18  41 

37.67 

1 18.58 

37.59 

18.74 

37.51 

18.90 

42 

43 

38.65 

18.85 

38.57 

! 19.02 

38.48 

19.19 

38.40 

19.35 

43 

44 

39.55 

19.29 

39.46 

19.46 

39.38 

19.63 

39.29 

19.80 

44 

45 

40.45 

19.73 

40.36 

19.90 

40.27 

20.08 

40. 18 

20.25 

45 

It 

I 41 .34 

20.17 

41.26 

! 20.35 

41.17 

20.53 

41.08 

20.70 

46 

47 

42.24 

20.60 

42.15 

20.79 

42.06 

20.97 

41.97 

21.15 

47 

48 

l 43. 14 

21  04 

43.05 

21.23 

42.96 

21.42 

42.86 

21.60 

- 48 

49 

1 44.04 

2.. 48 

43.95 

21.67 

43.85 

21.86 

43.76 

22.05 

49 

50 

44  .94 

21 .92 

44.84 

22.11 

44.75 

22.31 

44.65 

22.50 

50 

| 

' Dop. 

| Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

l 

6 

64  Deg 

63?  Deg. 

63  h Deg. 

684  Deg. 

! .a 

! Q 

I 

TRAVERSE  TABLE. 


125 


2 

Dog. 

m Deg. 

j m 

Deg. 

26» 

Deg. 

| S 

, | 

3 

O 

® 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

45.84 

22.36 

45.74 

22.56 

45764 

22776 

46  54 

22796 

64 

52 

46 . 74 

22.80 

46.64 

23.00 

46.54 

23.20 

46.43 

23.41 

52 

53 

47.64 

23.23 

47.53 

23.44 

47.43 

23.65 

47  33 

23.86 

53 

54 

48.53 

23.67 

48.43 

23.88 

48.33 

24-  09 

48  22 

24.31 

54 

55 

49.43 

24.11 

49.33 

24.33 

49.22 

24.54 

49.11 

24.76 

55 

56 

50.33 

24 . 55 

50.22 

24.77 

50.12 

24.99 

50.01 

25.21 

56 

57 

51.23 

24.99 

51.12 

25.21 

51.01 

25  43 

50.90 

25.66 

57 

58 

52.13 

25.43 

52.02 

25.65 

51.91 

25  88 

51.79 

26.11 

58 

59 

53.03 

25.86 

52 . 92 

26.09 

52.80 

26 . 33 

52.69 

26.56 

59 

GO 

53.93 

26.30 

53.81 

26 . 54 

53.70 

26  77 

53.58 

27.01 

60 

61 

54.83 

26.74 

54.71 

26 . 98 

54.59 

27.22 

54.47 

27.46 

61 

62 

55.73 

27. 18 

55.61 

27.42 

55.49 

27.66 

55.36 

27.91 

62 

63 

56 . 62 

27.62 

56 . 50 

27.86 

56.38 

28.11 

56.26 

28.36 

63 

64 

57  52 

28.06 

57.40 

28.31 

57.28 

28.56 

57. 15 

28.81 

64 

65 

58.42 

28.49 

58 . 30 

28.75 

58.17 

29.00 

58.04 

29.26 

65 

66 

59.32 

28.93 

59.19 

29.19 

59.07 

29.45 

58.94 

29.71 

66 

67 

60.22 

29.37 

60.09 

29.63 

59.96 

29.90 

59.83 

30. 16 

67 

63 

61.12 

29.81 

40.99 

30.08 

60.86 

30.34 

60.72 

30.61  ! 

68 

69 

62.02 

30.25 

61.88 

30.52 

61.75 

30.79 

61.62 

31.06 

69 

70 

62.92 

30.69 

62.78 

30.96 

62.65 

31.23 

62.51 

31.51 

70 

71 

63.81 

31.12 

63.63 

31.40 

63.54 

31.68 

63.40 

31.96 

71 

72 

64.71 

31.56 

64.57 

31.84 

64.44 

32.13 

64.29 

32.41 

72 

73 

65.61 

32 . 00 

65.47 

32.29 

65.33 

32.57 

65.  19 

32.86 

73 

74 

66.51 

32.44 

66.37 

32 . 73 

66.23 

33.02 

66.08 

33.31 

74 

75 

67.41 

32.88 

67.27 

33.17 

67.12 

33.46 

66.97 

33 . 76 

75 

76 

68.31 

33.32 

68.16 

33.61 

68.01 

33.91 

67.87 

34.21 

76 

77 

69.21 

33.75 

69.06 

34.06 

68.91 

34.36 

69.76 

34 . 66 

77 

78 

70.11 

34.19 

69.96 

34.50 

69.80 

34.80 

69.65 

35.11 

78 

T9 

71.00 

34.63 

70.85 

34.94 

70.70 

35.25 

70.55 

35.56 

79 

80 

71.90 

35.07 

71 .75 

35.38 

71 .59 

35.70 

71.44 

36.01 

80 

81 

72.80 

35.51 

72 . 65 

35.83 

72.49 

36. 14 

72.33 

36.46 

81 

82 

73.70 

35.95 

73 . 54 

36.27 

73.38 

36.59 

73.22 

! 36.91 

82 

83 

74.60 

36.38 

74.44 

36.71 

74.28 

37.03 

74.12 

1 37.36 

83 

84 

75.50 

36.82 

75.34 

37.15 

75. 17 

37.48 

75.01 

i 37.81 

84 

85 

76.40 

37.26 

76.23 

37.59 

76.07 

37.93 

75.90 

| 38.26 

85 

86 

77.30 

37.70 

77.13 

38.04 

76.96 

38.37 

76.80 

' 38.71 

86 

87 

78.20 

38.14 

78.03 

38  48 

77.86 

38.82 

77.69 

39.16 

87 

88 

' 79.09 

38.58 

78.92 

38.92 

78.75 

39.27 

78.58 

| 39.61 

88 

89 

79.99 

39.01 

79.82 

39.36 

79.65 

39.71 

79.48 

! 40.06 

89 

90 

| 80.89 

39.45 

80.72 

39.81 

80.54 

40.16 

80.37 

! 40.51 

90 

91 

81.79 

39.89 

81.62 

40.25 

81.44 

40.60 

81.26 

40.96 

91 

92 

82.69 

40.33 

82.51 

40.69 

82.33 

41.05 

82.15  41.41 

92 

93 

83.59 

40.77 

83.41 

41.13 

83.23 

41.50 

83.05 

41.86 

93 

94 

84  49 

41.21 

84.31 

41.58 

84.12 

41.94 

93.94 

42.31 

94 

95 

85.39 

41.65 

85 . 20 

42.02 

85.02 

42.39 

1 84.83 

42.76 

95 

96 

86.28 

42.08 

86.10 

42.46 

85.91 

1 42.83 

1 85.73 

43.21 

96 

97 

87.18 

42 . 52 

87.00 

42.90 

1 86.81 

: 43.28 

I 86.62 

43.66 

97 

98 

88.08 

42.96 

87.89 

43.34 

87.70 

43.73 

! 87.51 

44.11 

98 

99 

88.98 

43.40 

88.79 

43.79 

88.60 

44.17 

i 88.40 

44.56 

99 

100 

89.88 

43184_ 

89.68 

44.23 

89.49 

44.62 

89.30 

45.01 

100 

| 

Dop. 

| Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| Lat. 

l 

.3 

Q 

64  Deg. 

63?  Dog. 

631  Deg. 

63?  Deg. 

Q 

126 


I’HAVEKrSB  TABLE. 


g 

GO 

pT 

27  Deg. 

27}  Deg. 

27^ 

Deg. 

27} 

^eg. 

Z) 

f 

s 

1 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat  | 

Dep. 

D 

3 

i 

0.86 

0.45 

0.89 

0.46 

~0:89 

~L46 

’ 0.88  1 

0.47 

2 

1.78 

0.91 

1.78 

) . 12 

1.77 

0.92 

1 .77  1 

0.93 

i 

3 

2.67 

1.36 

2.67 

1.37 

2.66 

1.39 

V 65  1 

1.40 

3 

4 

3.56 

1 82 

3.56 

1.83 

3.55 

1.85 

0.54 

1.86 

i 

5 

4.45 

2 27 

4.46 

2.29 

4.44 

2.31 

4.42 

2.33 

5 

6 

5.35 

2.72 

5.33 

2.75 

6.32 

2.77 

5.31 

2.79 

(1 

7 

6.24 

3.18 

6.22 

3.21 

6.21 

3.23 

6.19 

3.26 

7 

8 

7.13 

3.63 

7.11 

3.66 

7.10 

3.69 

7.08 

3.72 

6 

9 

8.02 

4.09 

8.00 

4. 12 

7.98 

4.16 

7.96 

4.19 

9 

10 

8.91 

4.54 

8.89 

4.58 

8.87 

4 . 62 

8.85 

4.66 

10 

11 

9.80 

4.99 

9.78 

5.04 

9 . 76 

5 .08 

9.73 

5.12 

11 

12 

10.69 

5.45 

10.67 

5.49 

10.64 

5.54 

10.62 

5.59 

12 

13 

11.58 

5.90 

11.56 

5.95 

11.53 

6.00 

11.50 

6.05 

13 

14 

12.47 

6.36 

12.45 

6.41 

12.42 

6.46 

12.39 

6.52 

14 

15 

13.37 

6.81 

13.34 

6.87 

13.31 

6.93 

13.27 

6.98 

15 

10 

14.26 

7.26 

14.22 

7.33 

14.19 

7.39 

14.16 

7.45 

16 

1? 

15.15 

7.72 

15.11 

7.78 

15.08 

7.85 

15.04 

7.92 

17 

18 

16.04 

8.  17 

16.00 

8.24 

15.97 

8.31 

15.93 

8.38 

18 

19 

16.93 

8.63 

16.89 

8.70 

16.85 

8.77 

16.81 

8.85 

19 

20 

17.82 

9.08 

17.78 

9.16 

17.74 

9.23 

17.70  1 

9.31 

20 

21 

18.71 

9.53 

18.67 

9.62 

18.63 

9.70 

18.58 

~ 9.78 

21 

22 

19.60 

9.99 

19.56 

10.07 

19.51 

10.16 

19.47 

10.24 

22 

23 

20.49 

10.44 

20.45 

10.53 

20.40 

10.62 

20.35 

10.71 

23 

24 

21.38 

10.90 

21.34 

10.99 

21.29 

11.08 

21.24 

11.17 

24 

25 

22.28 

11.35 

22.23 

11.45 

22.18 

11.54 

22.12 

11.64 

25 

26 

23.17 

11.80 

23.11 

11.90  ; 

23.06 

12.01 

23.01 

12.11 

26 

27 

24.06 

12.26 

24.00 

12.36 

23.95 

12.47 

23.89 

12.57 

27 

28 

24.95 

12.71 

24.89 

12.82 

24.84 

12.93 

24.78 

13.04 

28 

29 

25.84 

13.17 

25.78 

13.28 

25.72 

13.39 

25.66 

13.50 

29 

30 

26.73 

13.62 

26.67 

13.74 

26.61 

13.85 

26.55 

13.97 

30 

31 

27.62 

14.07 

27.56 

14.19 

27.50 

14.31 

27.43 

14.43 

31 

32 

28.51 

14.53 

28.45 

14.65 

28.38 

14.78 

28.32 

14.90 

32 

33 

29.40 

14.98 

29.34 

15.11 

29.27 

15.24 

29.20 

15.37 

33 

34 

30.29 

15.44 

30.23 

15.57 

30.16 

15.70 

30.09 

15.83 

34 

35 

31.19 

15.89 

31.12 

16.03 

31.05 

16.16 

30.97 

16.30 

35 

36 

32.08 

16.34 

32.00 

16.48 

31.93 

16.62 

:l  ’ .86 

16.76 

36 

37 

32.97 

16.80 

32.89 

16.94 

32.82 

17.08 

•31.74 

17.23 

37 

38 

33.86 

17.25 

33.78 

17.40 

33.71 

17.55 

33.63 

17.69 

39 

39 

34.75 

17.71 

34.67 

17.86 

34.59 

18.01 

34.51 

18.16 

39 

4C 

35.64 

18.16 

35.56 

18.31 

35.48 

18.47 

35.40 

18.62 

40 

41 

36.53 

18.61 

36.45  18.77 

36.37 

18.93 

36.28 

19.09 

41 

42 

37.42 

19.07 

37.34 

19.23 

37.25 

19.39 

37.17 

19.56 

42 

43 

38.31 

1 19.52 

38.23 

19.69 

38.14 

19.86 

38.05 

20.02 

43 

44 

39.20 

i 1&  98 

39.12 

20.15 

39.03 

20.32 

38.94 

20  49 

44 

45 

! 40.10 

20.43 

40.01 

20.60 

39.92 

20.78 

39.82 

20  95 

45 

46 

40.99 

20.88 

40.89 

21.06 

40.80 

21.24 

40.71 

21.42 

46 

47 

1 41.88 

21.34 

41.78 

21.52  : 

41.69 

21.70 

41.59 

21.88 

47 

48 

42.77 

21.79 

42.67 

21.98 

42.58 

22.16 

4^.48 

22.35 

48 

49 

43.66 

22.25 

43.56 

22.44 

43.46 

22.63 

43.36 

22.82 

49 

60 

! 44.66 

22.70 

44.45 

22.89 

44.35 

23.03 

4Ar.25 

23  28 

50 

8 

g 

, Dep. 

! Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dop. 

I, at. 

0> 

o 

c 

5 

6 

1 

' 63  Deg. 

62}  Dog. 

621  Deg 

m Deg. 

1 

Q 

TRAVERSE  TA11LE. 


127 


o 

s* 

27  Dog. 

27{  Dog. 

271 

Deg. 

27 1 Deg. 

! o 

i S' 

ps 

a 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

n 

a 

61 

45.41 

23.15 

45.34  ! 23.35 

45.24 

23.55 

45.13 

23.75 

51 

55 

46.33 

23.61 

46.23 

23.81 

48.12 

24.01 

46.02 

24.21 

52 

53 

47.22 

24.  06 

47.12 

24.27 

47.01 

24.47 

46.90 

24.68 

53 

54 

48.11 

24.52 

48.01 

24.73 

47.90 

24.93 

47.79 

25.14 

51 

55 

49.01 

24.97 

48.90 

25.18 

48.79 

25.40 

48.67 

25.61 

55 

56 

49.90 

25.42 

49.78 

25.64 

49.67 

25.86 

49.56 

26.07 

56 

57 

50.79 

25.88 

50.67 

26.10 

50.56 

26.32 

50.44 

26.54 

57 

58 

51.68 

26.33 

51.56 

26.56 

51.45 

26.78 

51.33 

27.01 

58 

5S 

52.57 

26 . 79 

52.45 

27.01 

52.33 

27.24 

52.21 

27.47 

59 

60 

53.46 

27.24 

53.34 

27.47 

53.22 

27.70 

53.10 

27.94 

60 

61 

54.35 

27.69 

54.23 

27.93  1 

1 54.11 

28.17 

53.98 

28.40 

61 

62 

55.24 

28.15 

55.12 

28.39 

54.99 

28.63 

54.87 

28.87 

62 

63 

56.13 

28.60 

56.01 

28.85 

55.88 

29.09 

55.75 

29.33 

63 

64 

57.02 

29.06 

56.90 

29.30 

56.77 

29.55 

56.64 

29.80 

64 

65 

57  92 

29.51 

57.79 

29.76 

57.66 

30.01 

57.52 

30.26 

65 

66 

58.81 

29.96 

58.68 

30.22 

58.54 

30.48 

58.41 

30.73 

66 

67 

59.70 

30.42 

59 . 56 

30.68 

59.43 

30.94 

59.29 

31.20 

67 

68 

60.59 

30.87 

60.45 

31.14 

60.32 

31.40 

60.18 

31.66 

68 

69 

61.48 

31.33 

61.34 

31.59 

61.20 

31 .86 

61.06 

32.13 

69 

70 

62.37 

31.78 

62.23 

32.05 

62.09 

32.32 

61.95 

32.59 

70 

71 

63.26 

32.23 

63.12 

32.51 

62.98 

32 . 78 

62.83 

33.06 

'71 

72 

64.15 

32.69 

64.01 

32.97 

63.86 

33.25 

63.72 

33.52 

72 

73 

65.04 

33.14 

64.90 

33.42 

64.75 

33.71 

64.60 

33.99 

73 

74 

65.93 

33.60 

65.79 

33.88 

65.64 

34.17 

65.49 

34.46 

74 

75 

66.83 

34.05 

66.68 

34.34 

66.53 

34.63 

66.37 

34.92 

75 

76 

67.72 

34.50 

67.57 

34.80 

67.41 

35.09 

67.26 

35.39 

76 

77 

68.61 

34.96 

68.45 

35.26 

68.30 

35.55 

68.14 

35.85 

77 

78 

69.50 

35.41 

69.34 

35.71 

69.19 

36.02 

69.03 

36.32 

78 

79 

70.39 

35.87 

70.23 

36.17 

70.07 

36.48 

69.91 

36.78 

79 

80 

71.28 

36.32 

71-12 

36.63 

70.96 

36.94 

70.80 

37.25 

80 

81 

72.17 

36.77 

72.01 

37.09 

71.85 

37.40 

71.68 

37.71 

81 

82 

73.06 

37.23 

72.90 

37.55 

72.73 

37.86 

72.57 

38.18 

82 

83 

73.95 

37.68 

73.79 

38.00 

73.62 

38.33 

73.45 

38.65 

83 

84 

74.84 

38.14 

74.68 

38.46 

74.51 

38.79 

74.34 

39.11 

84 

85 

75.74 

38.59 

75.57 

38.92 

75.40 

39.25 

75.22 

39.58 

85 

86 

76.63 

39.04 

76.46 

39.38 

76.28 

39.71 

76.11 

40.04 

86 

87 

77.52 

39.50 

77.34 

39.83 

77.17 

40.17 

76.99 

40.51 

87 

88 

78  41 

39.95 

78.23 

40.29 

78.06 

40.63 

77.88 

40.97 

88 

89 

79.30 

40.41 

79.12 

40.75 

78.94 

41.10 

78.76 

41.44 

89 

90 

80.19 

10.86 

80.01 

41.21 

79.83 

41.56 

79.65 

41.91 

90 

91 

81.08 

41.31 

80.90 

41.67 

80.72 

42.02 

80.53 

42.37 

91 

92 

81  97 

41.77 

81.79 

42.12 

81.60 

42.48 

81.42 

42.84 1 

92 

93 

82  8C 

42.22 

82.68 

42.58 

82.49 

1 42.94 

82.30 

43.30 i 

93 

94  , 

83.75 

42.68 

83.57 

43.04 

83.38 

43.40 

83.19 

43.77 | 

94 

95  j 

84.65 

43.13 

84.46 

43.50 

84.27 

43.87 

84.07 

44.23  ; 

95 

9G 

85  54 

43.58 

85.35 

43.96 

85.15 

44.33  | 

84.96 

44.70 

96 

9?  | 

86.43 

44.04 

86.23 

44.41 

86.04 

44.79  ' 

85.84 

45.16 

97 

98 

87.32 

44.49 

87.12 

44.87 

86.93 

45.25 

86.73 

45.63 

98 

99  , 

88.21 

44.95 

88.01 

45.33 

87.81 

45.71 

87.61 

46.10 

99 

too 

89.10 

45^40 

88.90 

45.79 

88.70 

46.17 

88.50 

46.56 

100 

ST  1 

3 1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

5 

3 

1 j 

63  Deg. 

62|  Deg. 

621 

Deg. 

621  Deg. 

1 

Q 

128 


TRAVER8K  TABLE. 


a 

S’ 

p 

28  Deg. 

28$  Dog. 

1 28$  Deg. 

28$  Deg. 

1? 

P 

a 

» 

o 

L&t. 

Dep. 

L&t. 

Dep. 

Lat. 

Dep. 

Lat 

! Dep. 

3 

g 

1 

0.88 

0.47 

0.88 

0.47 

0.88 

~0748 

"Ik  86 

0.48 

1 

2 

1.77 

0.94 

1.76 

0.95 

1.76 

0.95 

1.75 

, 0.96 

3 

2.65 

1.41 

2.64 

1.42 

2.64 

1 .43 

2 . 63 

1 .44 

3 

4 

3.53 

1.88 

3.52 

1.89 

3.52 

1.91 

3.51 

1 .92 

4 

6 

4.41 

2.35 

4.40 

2.37 

4.39 

2.39 

4.38 

2.40 

5 

£ 

5.30 

2.82 

5.29 

2.84 

5.27 

2.86 

5.26 

2.89 

6 

? 

6.18 

3.29 

6.17 

3.31 

I 6.15 

3 . 34 

6.14 

3.37 

7 

8 

7.06 

3.76 

7.05 

3.79 

7.03 

3.82 

7.01 

3.85 

8 

9 

7.95 

4.23 

7.93 

4.26 

7.91 

4.29 

7.89 

4.33 

9 

10 

8.83 

4.69 

8.81 

4.73 

8.79 

4.77 

8.77 

4.81 

10 

11 

9.71 

5.16 

9.69 

5.21 

9.67 

5.25 

9.64 

5.29 

1 1 

12 

10.60 

5.63 

10.57 

5.68 

10.55 

5.73 

10.52 

5.77 

12 

13 

11.48 

6. 10 

11.45 

6.  15 

11.42 

6.20 

11.40 

6.25 

13 

14 

12.36 

6.57 

12.33 

6.63 

12.30 

6.68 

1 12.27 

6.73 

14 

15 

13.24 

7.04 

13.21 

7. 10 

13.18 

7.16 

i 13.15 

7.21 

15 

16 

14.  13 

7.51 

14.09 

7.57 

14.06 

7.63 

1 14.03 

7.70 

16 

17 

15.01 

7.98 

14.98 

8.05 

14.94 

8.11 

' 14.90 

8.18 

17 

18 

15.89 

8.45 

15.86 

8.52 

15.82 

8.59 

15.78 

8.66 

18 

19 

16.78 

8.92 

16.74 

8.99 

16.70 

9.07 

16.66 

9.14 

19 

20 

17.66 

9.39 

17.62 

9.47 

17.58 

9.54 

17.53 

9.62 

20 

21 

18.54 

9.86 

18.50 

9.94 

18.46 

10.02  1 

18.41 

10.10 

21 

22 

19.42 

10.33 

19.38 

10.41 

19.33 

10.50 

19.29 

10.58 

22 

23 

20.31 

10.80 

20.26 

10.89 

20.21 

10.97 

20.16 

11.06  , 

23 

24 

21.19 

11.27 

21.14 

11.36 

21.09 

11.45 

21.04 

11.54  | 

24 

25 

22.07 

11.74 

122.02 

11.83 

21.97 

11.93 

21.92 

12.02  1 

25 

26 

22.96 

12.21 

22.90 

12.31 

22.85 

12.41  i 

22.79 

12.51  1 

26 

27 

23.84 

12.68 

23.78 

12.78 

23.73 

12.88  1 

23.67 

12.99 

27 

28 

24.72 

13.15 

24.66 

13.25 

24.61 

13.36 

24.55 

13.47 

28 

29 

25.61 

13.61 

25.55 

13.73 

25.49 

13.84 

25.43 

13.95 

29 

30 

26.49 

14.08 

26.43 

14.20 

26.36 

14.31 

26.30 

14.43 

30 

31 

27.37 

14.55 

27.31 

14.67 

27.24 

14  79 

27.18 

14.91 

31 

32 

28.25 

15.02 

28.19 

15.15 

28.12 

15.27 

28.06 

15.39 

32 

33 

29.14 

15.49 

29.07 

15.62 

29.00 

15.7* 

28.93 

15.87 

33 

34 

30.02 

15.96 

29.95 

16.09 

29.88 

16.22 

29.81 

16.35 

34 

35 

30.90 

16.43 

30.83 

16.57 

30.76 

16.70 

30.69 

16.83 

35 

36 

31.79 

16.90 

31.71 

17.04 

31.64 

17.18 

31.56 

17.32 

36 

37 

32.67 

17.37 

32.59 

17.51 

32.52 

17.65 

32.44 

17.80 

37 

38 

33.55 

17.84 

33.47 

17.99 

33.39 

18.13 

33.32 

18.28 

38 

39 

34.43 

18.31 

34.35 

18.46 

34.27 

18.61 

34.19 

18  76 

39 

40 

35.32 

18.78 

35.24 

18.93 

35.15 

19.09 

35.07 

19.24 

40 

41 

36.20 

19.25 

36. 12 

19.41 

36.03 

19.56 

35.95 

19.72 

11 

42 

37.08 

19.72 

37.00 

19.88 

36.91 

20.04 

36.82 

20.20 

42 

43 

37.97 

20.19 

37.88 

20.35 

37.79 

20.52 

37.70 

20.68 

43 

44 

138  «5 

20.66 

38.76 

20.83 

38  67 

20.99 

38.58 

21.16 

14 

45 

1 39  73  1 

21.13 

39.64 

21 .30 

39.55 

21.47 

33.45 

21.64 

If 

4* 

40.62 

21 .60 

40.52 

21.77 

40.43 

21.95 

40.33 

22. 13 

1C 

47 

41.50  1 

22.07 

41.40 

22.25 

41.30 

22.43 

41.21 

22.61 

47 

4fc 

42.38 

22.53 

42.28 

22.72 

42.18 

22.90 

42.08 

23.09 

48 

49 

43 . 26 

23.00 

43.16 

23.19 

43.06 

23.38 

42.96 

23.57 

49 

50 

i 44.15 

23.47 

44.04 

23.67 

43.94 

23.86 

43.84 

24.05 

50 

8 

q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

L&t. 

Dep. 

Lat. 

6 

' L 

.2 

3 i 

62  Deg. 

61|  Deg. 

61$  Deg. 

61$  Deg. 

cd 

CD 

5 

TRAVEKSE  table. 


129 


cl 

m ; 

$T 

28  Deg. 

28*  Dog. 

28*  Deg. 

28*  Deg. 

O 

S’ 

r» 

eg 

5 

f i 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

3 

a 

a 

5’  ! 

52 

53 

54 
65 

56 

57 

58 

59 

60 

45.03 

45.91 

46.80 

47.68 

48.56 

49.45 

50.33 

51.21 

52.09 

52.98 

23.94 
24.41  : 
24.88  1 
25.35  | 
25.82  ! 
26.29 
26.76 
27.23 
27.70 
28.17 

44.93 

45.81 

46.69 

47.57 

48.45 

49.33 

50.21 

51.09 

51.97 

52.85 

24.14 

24.61 

25.09 

25.56 

26.03 

26.51 

26.98 

27.45 

27.93 

28.40 

44782" 

45.70 

46.58 

47.46 

48.33 

49.21 

50.09 

50.97 

51.85 

52.73 

24.34 

24.81 

25.29 

25.77 

26.24 

26.72 

27.20 

27.68 

28.15 

28.63 

44.71 

45.59 
46.47 
47.34 
48.22 
49.10 
49.97 
50.85 
51.73 

52.60 

24.53 

25.01 

25.49 

25.97 

26.45 

26.94 

27.42 

27.90 

28.38 

28.86 

2 

53 

51 

55 

56 

57 

58 

59 

60 

61 

53.86 

28.64 

53.73 

28.87 

53.61 

29.11 

53.48 

29.34 

61 

1 62 

54.74 

29.11 

54.62 

29.35 

54.49 

29.58 

54.36 

29.82 

62 

63 

55.63 

29.58 

55.50 

29.82 

55.37 

30.06 

55.23 

30.30 

63 

64 

56.51 

30.05 

56.38 

30.29 

56.24 

30 . 54 

56.11 

30.78 

64 

65 

57.39 

30.52 

57.26 

30.77 

57.12 

31.02 

56.99 

31.26 

65 

66 

58.27 

30.99 

58.14 

31.24 

58.00 

31.49 

57.86 

31.75 

66 

67 

59.16 

31.45 

59.02 

31.71 

58.88 

31.97 

58.74 

32.23 

67 

68 

60.04 

31.92 

59.90 

32.19 

59.76 

32.45 

59.62 

32.71 

68 

69 

60.92 

32.39 

60.78 

32.66 

60.64 

32.92 

60.49 

33.19 

69 

70 

61.81 

32.86 

61.66 

33.13 

61.52 

33.40 

61.37 

33.67 

70 

71 

62.69 

33.33 

62.54 

33.61 

62.40 

33.88 

62.25 

34 . 15 

71 

72 

63.57 

33.80 

63.42 

34.08 

63.27 

34.36 

63.12 

34.63 

72 

73 

64.46 

34.27 

64.30 

34.55 

64.15 

34.83 

64.00 

35.11 

73 

74 

65.34 

34.74 

65.19 

35.03 

65.03 

35.31 

64.88 

35.59 

74 

75 

66.22 

35.21 

66.07 

35.50 

65.91 

35.79 

65.75 

36.07 

75 

76 

67.10 

35.68 

66.95 

35.97 

66.79 

36.26 

66.63 

36.56 

76 

77 

67.99 

36.15 

67.83 

36.45 

67.67 

36.74 

67.51 

37.04 

77 

78 

j 68.87 

36.62 

68.71 

36.92 

68.55 

37.22 

68.38 

37.52 

78 

79 

69.75 

37.09 

69.59 

37.39 

69.43 

37.70 

69.26 

38.00 

79 

80 

70.64 

37.56 

70.47 

37.87 

70.31 

38.17 

70.14 

38.48 

80 

81 

71.52 

38.03 

71.35 

38.34 

71.18 

38.65 

71.01 

38.96 

81 

82 

72.40 

38.50 

72.23 

38.81 

72.06 

39.13 

71.89 

39.44 

82 

83 

73.28 

38.97 

73.11 

39.29 

72.94 

39.60 

72.77 

39.92 

83 

84 

74.17 

39.44 

73.99 

39.76 

73.82 

40.08 

73.64 

40.40 

84 

85 

75.05 

39.91 

74.88 

40.23 

74.70 

40.56 

74.52 

40.88 

85 

86 

75.93 

40.37 

75.76 

40.71 

75  58 

41.04 

75.40 

41.36 

86 

87 

j 76.82 

40.84 

76.64 

41.18 

76.46 

41.51 

76.28 

41.85 

87 

88 

77.70 

41.31 

77.52 

41.65 

77.34 

41.99 

77. 15 

42.33 

88 

89 

78.58 

41.78 

78.40 

42.13 

78.21 

142.47 

78.03 

42.81 

89 

90 

79.47 

42.25 

79.28 

42.60 

79.09 

42.94 

78.91 

43.29 

90 

o: 

80.35 

42 . 72 

80.16 

43.07 

79.97 

1 43.42 

79.7$ 

43.77 

91 

92 

81 .23 

43. 19 

81.04 

43.55 

80.85 

1 43.90 

80.66 

44  25 

92 

93 

I 82.11 

43.66 

81.92 

44.02 

81.73 

I 44.38 

81  .54 

44  73 

93 

94 

! 83.00 

44.13 

82.80 

44.49 

82.61 

! 44.85 

82.41 

4$.,°1 

91 

95 

j 83.88 

44.60 

83.68 

44.97 

l!  83.49 

45.33 

83.29 

45.  c9 

95 

96 

| 84.76 

1 45.07 

84.57 

45.44 

84.37 

45.81 

84.17 

! 46.17 

96 

97 

85.65 

45 . 54 

85.45 

45.91 

85.25 

16.28 

85.04 

' 16.66 

97 

98 

i 86.53 

46.01 

86  33 

46.39 

86.12 

46.76 

85.92 

47. 14 

98 

99 

87.41 

46.48 

87.21 

46.86 

87.00 

47.24 

86.80 

47.62 

99 

100 

88 . 29 

46.95 

88.09 

47.33 

87.88 

47.72 

87.67 

48.10 

lOO 

| 

Dep. 

J Lat. 

Dep. 

1 Lat. 

j Dep. 

1 

1 Lat. 

Dep. 

j Lat. 

© 

© 

C 

Q 

62  Deg. 

61 1 Deg. 

1 

61i  Deg 

61*  Deg. 

cd 

(U 

5 

130 


TRAVKR8K  TABLK. 


o 

55 

29  Dog. 

29*  Deg. 

29  i Deg. 

29*  Dog. 

C 

U) 

s 

5 

3 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  | 

Dep. 

Lat. 

Dep. 

5 

§ 

0.87 

0 48 

0.87 

0.49 

6787! 

~0.49 

~0 .87 

150 

1 

3 

1.75 

0.97 

1.74 

0.98 

1.74 

0.98 

1.74 

0.99 

2 

3 

2.62 

1.45 

2.62 

1.47 

2.61 

1.48 

2.60 

1.49 

3 

4 

3.50 

1.94 

3.49 

1.95 

3.48 

1.97 

3.47 

1.98 

4 

6 

4 37 

2.42 

4.  36 

2.44 

4.35 

2.46 

4.34 

2.48 

5 

6 

5.25 

2.91 

5.23 

2.93 

5.22 

2.95 

5.21 

2.98 

6 

7 

6.12 

3.39 

6.11 

3.42 

6.09 

3.45 

6.08 

3.47 

7 

8 

7.00 

3.88 

6.98 

3.91 

6.96 

3.94 

6.95 

3.97 

8 

9 

7.87 

4.36 

7.85 

4.40 

7.83 

4.43 

7.81 

4.47 

9 

10 

8.75 

4.85 

8.72 

4.89 

8.70 

4,92 

8.68 

4.96 

10 

11 

9.62 

5.33 

9.60 

5.37 

9.57 

5.42 

9.55 

5,46 

11 

12 

10.50 

5.82 

10.47 

6.86 

10.44 

5.91 

10.42 

5.95  ! 

12 

13 

11.37 

6.30 

11.34 

6.35 

11.31 

6.40 

11.29 

6.45 

13 

14 

12.24 

6.79 

12.21 

6.84 

12.18 

6.89 

12.15 

6.95 

14 

15 

13.12 

7.27 

13.09 

7.33 

13.06 

7.39 

13.02 

7.44 

15 

16 

13.99 

7.76 

13.96 

7.82 

13.93 

7.88 

13.89 

7.94 

16 

17 

14.87 

8.24 

14.83 

8.31 

14.80 

8.37 

14.76 

8.44 

17 

18 

15.74 

8.73 

15.70 

8.80 

15.67 

8.86 

15.63 

8.93 

18 

19 

16.62 

9.21 

16.58 

9.28 

16.54 

9.36 

16.50 

9.43 

19 

20 

17.49 

9.70 

17.45 

9.77 

17.41 

9.85 

17.36 

9.92 

20 

21 

18.37 

10. 18 

18.32 

10.26 

18.28 

10.34 

18.23 

10.42 

21 

22 

19.24 

10.67 

19. 19 

10.75 

19.15 

10.83 

19.10 

10.92 

22 

23 

20.12 

11.15 

20.07 

11.24 

20.02 

11.33 

19.97 

IV. 41 

23 

24 

20.99 

11.64 

20.94 

11.73 

20.89 

11.82 

20.84 

11.91 

24 

25 

21.87 

12.12 

21.81 

12.22 

21.76 

12.31 

21.70 

12.41 

25 

26 

22.74 

12.60 

22.68 

12.70 

22.63 

12.80 

22.57 

12.90 

26 

27 

23.61 

13.09 

23.56 

13.19 

23.50 

13.30 

23.44 

13.40 

27 

28 

24.49 

13.57 

24.43 

13.68 

24.37 

13.79 

24.31 

13.89 

28 

29 

25.36 

14.06 

25.30 

14.17 

25.24 

14.28 

25.18 

14.39 

29 

30 

26.24 

14.54 

26.17 

14.66 

26.11 

14.77 

26.05 

14.89 

30 

31 

27. 1 1 

15.03 

27.05 

15.15 

26.98 

15.27 

26.91 

15.38 

31 

32 

27.99 

15.51 

27.92 

15.64 

27.85 

15.76 

27.78 

15.88 

32 

33 

28.86 

16.00 

28.79 

16.12 

28.72 

16.25 

28.65 

16.38 

33 

34 

29.74 

16.48 

29.66 

16.61 

29.59 

16.74 

29.52 

16.87 

34 

36 

30.61 

16.97 

30.54 

17.10 

30.46 

17.23 

30.39 

17.37 

35 

36 

31.49 

17.45 

31.41 

17.59 

31.33 

17.73 

31.26 

17.86 

36 

37 

32.36 

17.94 

32.28 

18.08 

32.20 

18.22 

32.12 

18.36 

37 

38 

33.24 

18.42 

33.15 

18.57 

33.07 

18.71 

32.99 

18.86 

38 

39 

34.11 

18.91 

34.03 

19.06 

33.94 

19.20 

33.86 

19.35 

39 

40 

34.98 

19.39 

34.90 

19.54 

34.81 

19.70 

34.73 

19.85 

40 

41 

35.86 

19.88 

35.77 

20.03 

35.68 

20.19 

35.60 

20.34 

41 

42 

36.73 

20.36 

36.64 

20 . 52 

36.55 

20.68 

36.46 

20.84 

42 

43 

37.61 

20.85 

37.52 

21.01 

37.43 

21.17 

37.33 

21.34 

i 43 

44 

38.48 

21.33 

38.39 

21.50 

38.30 

21.67 

38.20 

21 .83 

! 44 

45 

39.36 

21 .82 

39.26 

21.99 

39.17 

22.16 

39.07 

22.33 

45 

46 

1 40.23 

22.30 

40.13 

22.48 

40.04 

22.65 

39.94 

22.83 

46 

47 

41.11 

22.79 

41.01 

22.97 

40.91 

23.14 

40.81 

23.32 

47 

48 

41.98 

23.27 

41.88 

23.45 

41.78 

23.68 

41.67 

23.82 

48 

49 

42  86 

23.76 

42.75 

23.94 

42.65 

24.13 

42.54 

24.31 

49 

60 

43.73 

24.24 

43.62 

24.43 

43.52 

24.62 

43.41 

24.81 

50 

8 

Dop. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dop. 

Lat. 

© 

o 

c 

b 

61  Dog. 

60f  Deg. 

60*  Dog. 

60*  Deg. 

1 

C 

TRAVERSE  TABLE. 


131 


g 

s* 

r+ 

29  Deg. 

29?  Deg. 

29  j 

Deg. 

29|  Deg. 

O 

55 

5 

Q 

« 

Lat.  1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 

8 

51 

44.61 

24.73 ' 

44.50 

24.92 

44739 

25.0 

44728 

25731 

51 

52 

45.48 

25.21 

45.37 

25.41 

45.26 

25.61 

45.15 

25.80 

52 

53 

46.35 

25.69 

46.24 

25.90 

46.13 

26.10 

46.01 

26.30 

63 

54 

47.23 

26.18 

47.11 

26.39 

47.00 

26.59 

46.88 

26.80 

54 

55 

48.10 

26 . 66 

47.99 

26.87 

47.87 

27.08 

47.75 

27.29 

55 

56 

48.98 

27.  15 

48.86 

27.36 

48.74 

27.58 

48.62 

27.79 

56; 

57 

49.85 

27.63 

49.73 

27.85 

49.61 

28.07 

49.49 

28.28 

57 

68 

50,73 

28.12 

50.60 

28.34 

50.48 

28.56 

50.36 

28.78 

58 

59 

51.60 

28.60 

51.48 

28.83 

51.35 

29.05 

51.22 

29.28 

59 

60 

52.48 

29.09 

52.35 

29.32 

52.22 

29.55 

52.09 

29.77 

60 

61 

53.35 

29.57 

53.22 

29.81 

53.09 

30.04 

52.96 

30.27 

61 

62 

54.23 

30.06 

54.09 

30.29 

53.96 

30.53 

53.83 

30.77 

62 

63 

55.10 

30.54 

54.97 

30.78 

54.83  l 

31.02 

54.70 

31.26 

63 

64 

55.98 

31.03 

55.84 

31.27 

55.70 

31.52 

55.56 

31.76 

64 

65 

56.85 

31  51 

56.71 

31.76 

56.57 

32.01 

56.43 

32.25 

65 

66 

57 . 72 

32.00 

57.58 

32.25 

57.44 

32.50 

57.30 

32.75 

66 

67 

58.00 

32.48 

58.46 

32.74 

58.31 

32.99 

58.17 

33.25 

67 

68 

59.47 

32.97 

59.33 

33.23 

59.18 

33.48 

59.04 

33.74 

68 

69 

60.35 

33.45 

60.20 

33.71 

60.05 

33.98 

59.91 

34.24 

69 

70 

61.22 

33.94 

61.07 

34.20 

60.92 

34.47 

60.77 

34.74 

70 

71 

62.10 

34.42 

61.95 

34.69 

61.80 

34.96 

61.64 

35.23 

71 

72 

62.97 

34.91 

62.82 

35.18 

62.67 

35.45 

62.51 

35.73 

72 

73 

63.85 

35.39 

63.69 

35.67 

63.54 

35.95 

63.38 

36.22 

73 

74 

64.72 

35.88 

64.56 

36.16 

64.41 

36.44 

64.25 

36.72 

74 

75 

65.60 

36.36 

65.44 

36.65 

65.28 

36.93 

65.11 

37.22 

75 

76 

66.47 

36.85 

66.31 

37.14 

66.15 

37.42 

65.98 

37.71 

76 

77 

67.35 

37.33 

67.18 

37.62 

67.02 

37.92 

66.85 

38.21 

77 

78 

68.22 

37.82 

68.05 

38.11 

67.89 

38.41 

67.72 

38.70 

78 

79 

69.09 

38.30 

68.93 

38.60 

68.76 

38.90 

68.59 

39.20 

79 

80 

69.97 

38.78 

69.80 

39.09 

69.63 

39.39 

69.46 

39.70 

80 

81 

70.84 

39.27 

70.67 

39.58" 

70.50 

39.89 

70.32 

40.19 

81 

82 

71.72 

39.75 

71.54 

40.07 

71.37 

40.38 

71.19 

40.69 

82 

83 

72.59 

40.24 

72.42 

40.56 

72.24 

40.87 

72.06 

41.19 

83 

84 

73.47 

40.72 

73.29 

41.04 

73.11 

41.36 

72.93 

41.68 

84 

85 

74.34 

41.21 

74.16 

41.53 

73.98 

41.86 

73.80 

42.18 

85 

86 

75.22 

41.69 

75.03 

42.02 

74.85 

42.35 

74.67 

42.67 

86 

87 

76.09 

42.18 

75.91 

42.51 

75.72 

42.84 

75.53 

43.17 

1 87 

88 

76.97 

42.66 

76.78 

43.00 

76.59 

43.33 

76.40 

43.67 

38 

89 

77.84 

43.15 

77.65 

43.49 

77.46 

43.83 

77.27 

44.16 

89  , 

90 

78.72 

43.63 

78.52 

43.98 

78.33 

44.32 

78.14 

44.66  | 90  1 

91 

79.59 

44.12 

79.40 

44.46 

79.20 

44.81 

79.01 

45.16 

91 

92 

80.46 

44.60 

80.27 

44.95 

80.07 

45.30 

79.87 

45.65 

; 95 

93 

81.34 

45.09 

81.14 

45.44 

80.94 

45.80 

80.74 

46.15 

03 

94 

1 82.21 

45.57 

82.01 

45.93 

81.81 

46.29 

81.61 

46.64 

91 

95 

83.09 

46.06 

82.89 

46.42 

82.68 

46.78 

82.48 

47.14 

95 

96 

83.96 

46 . 54 

83.76 

46.91 

83.55 

47.27 

83.35 

47.  64 

j 90 

97 

i 84.84 

47.03 

84.63 

47.40 

84.42 

47.77 

84.22 

48  13 

97 

98 

! 85.71 

47.51 

85.50 

47.88 

85.29 

48.26 

85.08 

48 . 63  1 98 

99 

j 86.59 

1 48.00 

86.38 

48.3? 

86.17 

48.75 

85.95 

49.13 

99 

100 

87.46 

| 48.48 

87.25 

48.86 

■ 87.04 

49.24 

86.82 

49.62 

100 

| 

Dep. 

! Lai.. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

a 

1 

O 

61  Deg. 

60?  Deg. 

601 

Deg. 

60?  Deg. 

ad 

00 

lS 

132 


ritA  VERSE  TAfllE 


V 

1* 

p> 

30  Deg. 

m Dog. 

30^  Dog. 

30|  Deg. 

O 

JD 

3 

n 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep.  j 

D 

3 

1 

0.87 

~Oo 

o.sT 

0.50 

J.86 

0.51 

0.86“ 

OTsi  j 

l 

2 

1.73 

1.00 

1.73 

1.01 

1.72 

' .02 

1.72  ; 

1 .02  I 

2 

3 

2.60 

1.50 

2 59 

1.51 

2.58  1 

1.52  j 

2 58 

1.53 

i 

4 

3.46 

2.00 

3.46 

2.02 

3.45 

2.03  1 

3 44 

2.05  | 

4 

5 

4.33 

2.50 

4.32 

2 . 52 

4.31 

2 . 54 

4 30 

2.56 

5 

6 

5.20 

3.00 

5.18 

3.02 

5.17 

3.05 

5.16 

3.07 

S 

7 

6.06 

3.50 

6.05 

3.53 

6.03 

3.55 

6.02 

3.58 

1 

8 

6.93 

4.00 

6.91 

4.03 

6.89 

4.06 

| 6.88 

4.09 

8 

9 

7.79 

4.50 

7.77 

4.53 

7.75 

4.57 

7.73 

4.60 

9 

10 

8.66 

5.00 

8.64 

5.04 

8.62 

5.08 

8.59 

5.11 

10 

11 

9.53 

5.50 

9.50 

5.54  ! 

9.48 

5.58 

9.45 

5.62  | 

; • 

12 

10.39 

6.00 

10.37 

6.05 

10.34 

6.09 

10.31 

6.14 

\2 

13 

11.26 

6.50 

11.23 

6.55 

11.20 

6.60 

11.17 

6.65 

13 

14 

12.12 

7.00 

12.09 

7.05 

12.06 

7.11 

12.03 

7. 16 

14 

15 

12.99 

7.50 

12.96 

7.56 

12.92 

7.61 

12.89 

7.67 

15 

16 

13.86 

8.00 

13.82 

8.06 

13.79 

8.12 

13.75 

8.18 

16 

17 

14.72 

8.50 

14.69 

8.56 

14.65 

8.63 

14.61 

8.69 

17 

18 

15.59 

9.00 

15.55 

9.07 

15.51 

9.14 

15.47 

9.20 

18 

19 

16.45 

9.50 

16.41 

9.57  ; 

16.37 

9.64 

16.33 

9.71 

19 

20 

17.32 

10.00 

17.28 

10.08 

17.23 

10.15 

17.15 

10.23 

20 

21 

18.19 

10.50 

18.14 

10.58  | 

18.09 

10.66 

18.05 

10.74 

21 

22 

19.05 

11.00 

19.00 

11.08 

18.96 

11.17 

18.91 

11.25 

22 

23 

19.92 

11.50 

19.87 

11.59  | 

19.82 

11.67 

19.77 

11.76 

23 

24 

20.78 

12.00 

20.73 

12.09 

20.68 

12. 18 

20.63 

12.27 

24 

25 

21 .65 

12.50 

21.60 

12.59 

21.54 

12.69 

21.49 

12.78 

25 

26 

22.52 

13.00 

22.46 

13.10 

22.40 

13.20 

22.34 

13.29 

26 

27 

23.38 

13.50 

23.32 

13.60 

23.26 

13.70 

23.20 

13.80 

27 

28 

24.25 

14.00 

24.19 

14.11 

24.13 

14.21 

24.06 

14.32 

28 

29 

i 25.11 

14.50 

25.05 

14.61 

24.99 

14.72 

24.92 

14.83 

29 

30 

! 25.98 

15.00 

25.92 

15.11 

25.85 

15.23 

25.78 

15.34 

30 

31 

26.85 

15.50 

26.78 

15.62 

26.71 

15.73 

26.64 

15.85 

31 

32 

1 27.71 

16.00 

27.64 

16.12 

27.57 

16.24 

27.50 

16.36 

32 

33 

28.58 

16.50 

28.51 

16.62 

28.43 

16.75 

28.36 

16.87 

33 

34 

, 29.44 

17.00 

29.37 

17.13 

29.30 

17.26 

29.22 

17.38 

34 

35 

! 30.31 

17.50 

30.23 

17.63 

30.16 

17.76 

30.08 

17.90 

35 

36 

31.18 

18.00 

31.10 

18.14 

31.02 

18.27 

30.94 

18.41 

36 

37 

32.04 

18.50 

31.96 

18.64 

31.88 

18.78 

31.80 

18.92 

37 

38 

32.91 

19.00 

32.83 

19.14 

32.74 

| 19.29 

32.66 

19.43 

38 

39 

33.77 

19.50 

33.69 

19.65 

33.60 

19.79 

33.52 

19.94 

39 

40 

34.64  20.00 

34.55 

20.15 

34.47 

20.30 

34.38 

20.45 

40 

41 

35.51 

20.50 

35.42 

20.65 

35.33 

20.81 

35.24 

20.96 

'41 

42 

36.37 

21 .00 

36 . 28 

21.16 

36.19 

21.32 

36.  10 

21.47 

42 

43 

37.24 

21.50 

37  14 

21.66 

37.05 

21.82 

36.95 

21.99 

' 13 

44 

38.11 

22.00 

38.01 

22.17 

37.91 

22.33 

37.81 

22.50 

44 

45 

38.97 

22.50 

38.87 

22.67 

38 . 77 

22 . 84 

38.67  | 23.01 

15 

46 

39.84 

23.00 

39.74 

23.17 

39.63 

23.35 

39.53 

23.52 

ie 

4* 

40.70 

23.50 

40.60 

1 23.68 

40.50 

23.85 

40.39 

j 24.03 

17 

48 

,41.57 

24.00 

41.46 

24.18 

41  36 

24.36 

41 .25 

i 24.54 

18 

49 

42.44 

24 . 50 

42.33 

24 . 68 

42.22 

24.87 

42.11 

25.05 

4S 

50 

43.30 

25.00 

43.19^ 

25. 19 

43.08 

i 25.38  j 

42.97 

25.56 

60 

8 

i 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

j Lat. 

Dep. 

j Lat. 

® 

o 

a 

3 

.a 

a 

60  Dog 

69J  Deg. 

59  \ Deg. 

59}  Deg. 

2 

a o 

s 

TKAVEKSE  TABLE. 


133 


9 

30  Deg. 

304  Deg. 

30*  Deg. 

30 1 Deg. 

i % 

I 

3 

n 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

r. 

(C 

51 

44.17 

25.50 

44.06 

25 . 69 

43.94 

25.88 

43783* 

*26TW 

~51 

52 

45.03 

26.00 

44.92 

26.20 

44.80 

26.39 

44.69 

26.59 

52 

53 

45.90 

26  50 

45.78 

26.70 

45.67 

26.90 

45.55 

27.10 

53 

54 

46.77 

27  00 

46.65 

27.20 

46.53 

27.41 

46.41 

27.61 

I 54 

55 

47.63 

27.50 

47.51 

27.71 

47.39 

27.91 

47.27 

28.12 

55 

56 

48.50 

28.00 

48.37 

28.21 

48.25 

28.42 

48.13 

28.63 

66 

57 

49.36 

28.50 

49.24 

28 . 72 

49.11 

28  98 

48.99 

'29.14 

57 

58 

50  23 

29.00 

50.10 

29.22 

49.97 

29.44 

49.85 

29.65 

58 

59 

51.10 

29.50 

50.97 

29.72 

50.84 

29.94 

50.70 

30.17 

59 

60 

51.96 

30. 00 

51.83 

30.23 

51.70 

30.45 

51.56 

30.68 

60 

61 

52.83 

30.50 

52.69 

30.73 

52.56 

30.96 

52.42 

31 . 19 

61 

62 

53.69 

31.00 

53.56 

31.23 

53.42 

31.47 

53.28 

31.70 

62 

63 

54.56 

31.50 

54.42 

31.74 

54.28 

31.97 

54.14 

32.21 

63 

64 

55.43 

32.00 

55.29 

32.24 

55.14 

32.48 

55.00 

32.72 

64 

65 

56.29 

32.50 

56.15 

32.75 

56.01 

32.99 

55.86 

33.23 

65 

66 

57.16 

33.00 

57.01 

33.25 

56.87 

33.50 

56.72 

33.75 

66 

67 

58.02 

33.50 

57.88 

33.75 

57.73 

34.01 

57.58 

34.26 

67 

68 

58.89 

34.00 

58.74 

34.26 

58.59 

34.51 

58.44 

34.77 

68 

69 

59.76 

34.50 

59.60 

34.76 

59.45 

35.02 

59.30 

35.28 

69 

70 

60.62 

35.00 

60.47 

35.26 

60.31 

35.53 

60.16 

35.79 

70 

71 

61.49 

35.50 

61.33 

35.77 

61.18 

36.04 

61.02 

36.30 

71 

72 

62.35 

36.00 

62.20 

36.27 

62.04 

36.54 

61.88 

36.81 

72 

73 

63.22 

36.50 

63.06 

36.78 

62.90 

37.05 

62.74 

37.32 

73 

74 

64.09 

37.00 

63.92 

37.28 

63.76 

37.56 

63.60 

37.84 

74 

75 

64.95 

37.50 

64.79 

37.78 

64.62 

38.07 

64.46 

38.35 

75 

76 

65.82 

38.00 

65.65 

38.29 

65.48 

38.57 

65.31 

38.86 

76 

77 

66.68 

38.50 

66.52 

38.79 

66.35 

39.08 

66.17 

39.37 

77 

78 

67.55 

39.00 

67.38 

39.29 

67.21 

39.59 

67.03 

39.88 

78 

79 

68.42 

39.50 

68.24 

39.80 

68.07 

40.10 

67.89  1 

40.39 

79 

80 

69.28 

40.00 

69.11 

40.30 

68.93 

40.60 

68.75  1 40.90 

80 

81 

70.15 

40.50 

69 . 97 

40.81 

69.79 

41.11 

69.61 

41.41 

81 

82 

71.01 

41.00 

70.83 

41.31 

70.65 

41.62 

70.47 

41.93 

82 

83 

71.88 

41.50 

71.70 

41.81 

71.52 

42.13 

71.33 

42.44 

83 

84 

72.75 

42.00 

72.56 

42.32 

72.38 

42.63 

72.19 

42.95 

84 

85 

73.61 

42.50 

73.43 

42.82 

73.24 

43.14 

73.05 

43.46 

85 

86 

74.48 

43.00 

74.29 

43.32 

74.10 

43.65 

73.91 

43.97 

86 

87 

75.34 

43.50 

75.15 

43.83 

74.96 

44.16 

74.77 

44.48 

87 

88 

76.21 

44.00 

76.02 

44.33 

75.82 

44.66 

75.63 

44.99 

88 

89 

77.08 

44.50 

76.88 

44.84 

76.68 

45.17 

76.49 

45.51 

89 

90 

77.94 

45.00 

77.75 

45.34 

77.55 

45.68 

77.35 

46.02 

90 

91 

78.81 

45.50 

78.61 

45.84 

78.41 

46.19 

78.21 

46.53 

91 

92 

79.67 

46.00 

79.47 

46.35 

79.27 

46.69 

79.07 

47.04 

92 

93 

80.54 

46.50 

80.34 

46.85 

80. 13 

47.20 

79.92 

47.55 

93 

94 

i 81.41 

47.00 

81.20 

47.35 

80.99 

47.71 

80.78 

48.06 

94 

95 

82.27 

47.50 

82.06 

47.86 

81.85 

48.22 

81.64 

48.57 

95 

96 

1 83.14 

48.00 

82.93 

48.36 

82.72 

48.72 

82.50 

49.08 

96 

97  I 84.00 

48.50 

83.79 

48.87 

83.58 

49.23 

83.36 

49.60 

97 

98 

i 84.87 

49.00 

84.66 

49.37 

84.44 

49.74 

84.22 

50.11 

98 

99 

85.74 

49.50 

85.52 

49.87 

85.30 

50.25 

85.08 

50.62 

99 

100 

86.60 

50. 00 

86.38 

50.38 

86.16 

50.75 

85.94 

51.13 

100 

8 

a 

Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0} 

o 

c 

0) 

W 

.3 

60  Deg. 

1 

59|  Deg. 

594  Dog. 

59|  Deg. 

ci 

75 

c 

134 


TltAVEKiSK  TAHi.fi 


Distance.! 

31  Deg. 

Mi  Dog. 

31  i Deg. 

31*  Dog.  i 

Distance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.86 

0. 

51 

0.85 

0. 

52 

0.85 

0. 

52 

0 .85 

0, 

.53 

1 

2 

1.71 

1. 

03 

1.71 

1. 

04 

1.71 

1. 

04 

1.70 

1, 

.05 

2 

3 

2.57 

1 

55 

2.56 

1. 

56 

2.66 

1. 

57 

2.55 

1, 

,58 

3 

4 

3.43 

2. 

06 

3.42 

2. 

08 

3.41 

2. 

09 

3.40 

2, 

.10 

4 

5 

4.29 

2. 

58 

4.27 

2. 

59 

4.26 

2. 

61 

4.25 

2, 

.63 

6 

e 

6. 14 

3. 

09 

5.13 

3. 

11 

5.12 

3. 

13 

5.10 

3 

.16 

6 

7 

6.00 

3. 

61 

5.98 

3. 

63 

5.97 

3. 

66 

5.95 

3, 

,68 

7 

8 

6.86 

4. 

12 

6.84 

4. 

15 

6.82 

4. 

18 

6.80 

4, 

,21 

8 

9 

7.71 

4. 

64 

7.69 

4. 

67 

7.67 

. 4. 

70 

7.65 

4, 

,74 

9 

10 

8.57 

5. 

15 

8.55 

5. 

19 

8.53  - 

5. 

22 

8.50 

5, 

.26 

10 

11 

9 43 

5. 

67 

9.40 

5. 

71 

9.38 

5. 

75 

9.35 

5, 

.79 

11 

12 

10.29 

6. 

18 

10.26 

6. 

23 

10.23 

6. 

27 

10.20 

6. 

.31 

12 

13 

11.14 

6. 

70 

11.11 

6. 

74 

11.08  , 

6. 

79 

11.05 

6, 

.84 

13 

14 

12.00 

7. 

21 

11.97 

7. 

26 

11.94  | 

7. 

31 

11.90 

7, 

.37 

14 

15 

12.86 

7. 

73 

12.82 

7. 

78 

12.79 

7. 

,84 

12.76 

7. 

.89 

1 '5 

16 

13.71 

8. 

24 

13.68 

8. 

30 

13.64 

8. 

36 

13.61 

8. 

,42 

16 

17 

14.57 

8. 

76 

14.53 

8. 

82 

14.49 

8. 

88 

14.46 

8, 

.95 

17 

18 

15.43 

9. 

27 

15.39 

9. 

34 

15.35 

9. 

40 

15.31 

9, 

.47 

18 

19 

16.29 

9. 

79 

16.24 

9. 

,86 

16.20 

9. 

93 

16.16 

10, 

.00 

19 

20 

17.14 

10. 

30 

17.10 

10. 

38 

17.05 

10. 

,45 

17.01 

10, 

.52 

20 

21 

18.00 

10. 

82 

17.95 

10. 

,89 

17.91 

10. 

,97 

17.86 

11, 

.05 

21 

22 

18.86 

11 

33 

18.81 

11. 

,41 

18.76 

11. 

,49 

18.71 

11, 

.58 

22 

23 

19.71 

11. 

.85 

19.66 

11, 

,93 

19.61 

12. 

,02 

19.56 

12, 

.10 

23 

24 

20.57 

12. 

36 

20.52 

12, 

,45  | 

20.46 

12. 

,54 

20.41 

12, 

.63 

24 

25 

21.43 

12. 

,88 

21.37 

12. 

,97  | 

21.32 

13. 

,06 

1 21.26 

13, 

. 16 

25 

26 

22.29 

13. 

,39 

22.23 

13. 

,49  i 

22.17 

13. 

,58 

,22.11 

13, 

.68 

26 

27 

23.14 

13. 

,91 

23.08 

14. 

,01  i 

23.02 

14. 

,11 

22.96 

14, 

.21 

27 

28 

24.00 

14. 

,42 

23.94 

14. 

,53  i 

23.87 

14, 

,63 

23.81 

14, 

.73 

28 

29 

124.86 

14. 

,94 

24.79 

15. 

,04  ! 

24.73 

15, 

,15 

24.66 

15, 

.26 

29 

30 

i 25.71 

15. 

,45 

25.65 

15. 

,56 

25.58 

15, 

,67 

25.51 

15, 

.79 

30 

31 

26.57 

15. 

,97 

26.50 

16. 

.08 

26.43 

16. 

,20 

26.36 

18, 

.31 

31 

32 

27.43 

16. 

,48 

27.36 

16, 

,60 

27.28 

16. 

,72 

27.21 

16, 

.84 

32 

33 

28.29 

17. 

,00 

28.21 

17. 

, 12 

28.14 

17. 

,24 

28.06 

17, 

.37 

33 

34 

29.14 

17. 

,51 

29.07 

17, 

.64 

28.99 

17. 

,76 

28.91 

17, 

.89 

34 

35  1 30.00 

18. 

,03 

29.92 

18, 

.16 

29.84 

18. 

,29 

29.76 

18, 

.42 

35 

36 

30.86 

18, 

,54 

30.78 

18, 

.68 

30.70 

18. 

,81 

30.61 

18, 

.94 

36 

37 

31.72 

19. 

,06 

31.63 

19, 

.19 

31.55 

19. 

.33 

31.46 

19, 

.47 

37 

38 

32.57 

19. 

,57 

32.49 

19, 

.71 

32.40 

19. 

,85 

32.31 

20, 

00 

38 

39 

33.43 

20, 

.09 

33.34 

20, 

.23 

33.25 

20. 

,38 

33.16 

20 

52 

39 

40 

34.29 

1 20, 

,60 

34.20 

20, 

.75 

34.11 

20, 

,90 

34.01 

21 

05 

40 

41 

35.14 

21, 

.12 

35.05 

21, 

.27 

34.96 

21. 

,42 

34.86 

21, 

.57 

41 

42 

36.00 

21, 

.63 

35.91 

21. 

.79 

35.81 

21, 

,94 

35.71 

22, 

. 10 

42 

43 

36.86 

22, 

.15 

36.76 

22, 

,31 

36.66 

22, 

.47 

36.57 

22, 

.63  1 

43 

44 

! 37.72 

22. 

,66 

37.62 

22, 

.83 

37.52 

22. 

,99 

37.42 

, 23, 

.15  , 

44 

45 

38.57 

23, 

,18 

38.47 

23. 

.34 

38.37 

23. 

,51 

38.27 

23 

.68 

45 

46 

39.43 

23. 

,69 

39.33 

23. 

,86 

39.22 

24. 

,03 

39.12 

24, 

.21 

1 46 

17 

40.29 

24. 

.21 

40. 18 

24. 

,38 

40.07 

24. 

,56 

39.97 

24, 

.73 

, 47 

48 

1 41.14 

24, 

,72 

41  04 

24. 

,90 

40.93 

25. 

,08 

40.82 

25 

.26  j 

! 48 

49 

42.00 

25, 

,24 

41.89 

25, 

.42 

41.78 

25. 

,60 

41.67 

25, 

.78 

49 

50 

| 42.86 

25. 

,75 

42.75 

25. 

.94 

42.63 

26. 

12 

42.52 

26, 

.31 

, 50 

8 

s 

1 

Q 

Dep. 

| Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 © 

1 § 

!.! 

59  Deg. 

1 

58*  Deg. 

58J  Deg. 

58*  P*jg. 

TRAVBR8E  TABLE 


135 


o 

so' 

p 

31  Deg. 

31$  Deg. 

311 

Deg. 

31$  Deg. 

D 

05 

P 

a 

o 

o 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  1 

Dep. 

3 

8 

61 

43.72 

26.27 

43.60 

26.46 

43.48 

26.65 

43.37 

■26784'  ! 

51 

62 

44.57 

26.78 

44.46 

26.98 

44.34 

27.17 

44.22 

27.36 

52 

63 

45.43 

27.30 

45.31 

27.49 

45.19 

27.69 

45.07 

27.89  1 

53 

54 

46.29 

27.81 

46.17 

28.01 

46.04 

28.21 

45.92  ! 28.42  j 

54 

55 

47.14 

28.33 

47.02 

28.53 

46.90 

28.74 

46. ^7  j 28.94  1 

55 

56 

48.00 

28.84 

47.88 

29.05 

47.75 

29.26 

47.62  i 

29.47  | 

56 

6? 

48.86 

29.36 

48.73 

29.57 

48.60 

29.78 

48.47 

29.99 

57 

58  1 49.72 

29.87 

49  58 

30.09 

49.45 

30.30 

49.32 

30.52 

58 

59 

50.57 

30.39 

50  44 

30.61 

50.31 

30.83 

50.17 

31.05 

59 

60 

51.43 

30.90 

51.29 

31.13 

51.16 

31 .35 

51.02 

31.57 

60 

61 

52.29 

31.42 

52.15 

31.65 

52.01 

31 .87 

51.87 

32.10 

61 

62 

53.14 

31.93 

53.00 

32.16 

52.86 

32.39 

52.72 

32.63 

62 

63 

54.00 

32.45 

53.86 

32.68 

53.72 

32.92 

53.57 

33.15 

63 

64 

54.86 

32.96 

54.71 

33.20 

54.57 

33.44 

54.42 

33.68 

64 

65 

55.72 

33.48 

55.57 

33.72 

55.42 

33.96 

55.27 

34.20 

65 

66 

56.57 

33.99 

56.42 

34.24 

56.27 

34.48 

56.12 

34.73 

66 

67 

57.43 

34.51 

57.28 

34.76 

57.13 

35.01 

56.98 

35.26 

67 

68 

58.29 

35.02 

58.13 

35.28 

57.98 

35.53 

57.82 

35.78 

68 

69 

59.14 

35.54 

58.99 

35.80 

58.83 

36.05 

58.67 

36.31 

69 

70 

60.00 

36.05 

59.84 

36.31 

59.68 

36.57 

59.52 

36.83 

70 

71 

60.86 

36.57 

60.70 

36.83 

"60.54 

37.10 

60.37 

37.36 

71 

72 

61.72 

37.08 

61.55 

37.35 

61.39 

37.62 

61.23 

37.89 

72 

73  I 

62.57 

37.60 

62.41 

37.87 

62.24 

38.14 

62.08 

38.41 

73 

74 

63.43 

38.11 

63.26 

38.39 

63.10 

38.66 

62.93 

38.94 

74 

75 

64.29 

38.63 

64.12 

38.91 

63.95 

39.19 

63.78 

39.47 

75 

76 

65.14 

39.14 

64.97 

39.43 

64.80 

39.71 

64.63 

39.99 

76 

77 

66.00 

39.66 

65.83 

39.95 

65.65 

40.23 

65.48 

40.52 

77 

78 

66.86 

40.17 

66.68 

40.46 

66.51 

40.75 

66.33 

41.04 

78 

79 

67.72 

40.69 

67.54 

40.98 

67.36 

41.28 

67.18 

41.57 

79 

80 

68.57 

41.20 

68.39 

41.50 

68.21 

41.80 

68.03 

42.10 

80 

81 

69.43 

41 .72 

69.25 

42.02 

69.06 

42.32 

68.88 

42.62 

81 

82 

70.29 

42.23 

70.10 

42.54 

69.92 

42.84 

69.73 

43.15 

82 

83 

71.14 

42.75 

70.96 

43.06 

70.77 

43.37 

70.58 

43.68 

83 

84 

72.00 

43.26 

71.81 

43.58 

71.62 

43.89 

71.43 

44.20 

84 

85 

72.86 

43.78 

72.67 

44.10 

72.47 

44.41 

72.28 

44.73 

85 

86 

73.72 

44.29 

73.52 

44.61 

73.33 

44.93 

73.13 

45.25 

! 86 

87 

74.57 

44.81 

74.38 

45.13 

74.18 

45.46 

73.98 

45.78 

87 

88 

75.43 

45.32 

75.23 

45.65 

75.03 

45.98 

74.83 

46.31 

88 

89 

76 . 29 

45.84 

76.09 

46.17 

75.88 

46.50 

75.68 

46.83 

89 

90 

77.15 

46.35 

76.94 

46.69 

76.74 

47.02 

76.53 

47.36 

90 

91 

78.00 

46.87 

77.80 

47.21 

77.59 

47.55 

77.38 

47.89 

91 

92 

78.86 

47.38 

78.65 

47.73 

78.44 

48.07 

78.23 

48.41 

92 

93 

79.72 

47.90 

79.51 

48.25 

79.30 

48.59 

79.08 

48.94 

93 

94 

80.57 

48.41 

80.36 

48.76 

80.15 

49.11 

79  93 

49.47 

91 

95 

81.43 

48.93 

81.22 

49.28 

81.00 

49.64 

80  78 

49.99 

95 

96 

82.25 

j 49.44 

82.07 

49.80 

81.85 

50.16 

81.63 

50.52 

96 

y7 

83.15 

1 49.96 

82.93 

00.32 

82.71 

50  .68 

| 82.48 

51.04 

97 

98 

84.00 

! 50.47 

83.78 

50.84 

83.56 

51.20 

83.33 

I 51.57 

98 

99 

! 84.86 

i 50.99 

84.64 

51.36 

84.41 

51.73 

84.18 

52.10 

99 

.00 

' 85.72 

! 51.50 

85.49 

51.88 

85.26 

52.25 

85.04 

52.62 

100 

i 

a 

I Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

s 

G 

a 

m 

5 

1 

1 59  Deg. 

58^  Deg. 

58^ 

Deg. 

58$  Deg. 

ed 

on 

ifJ 

TRAVERSE  TABLE. 


130 


u 

K* 

ST 

32  Deg. 

32i  Deg. 

32£  Deg. 

32$  Deg. 

5 

5>* 

P 

1 

© 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

0 

O 

ct> 

1 

0.85 

0.53 

0.85 

0.53 

0.84 

0.54 

0.84 

0.54 

1 

2 

1.70 

1.06 

1.69 

1.07 

1.69 

1.07 

1.68 

1.08 

2 

3 

2.54 

1.59 

2.54 

1.60 

2.53 

1.61 

2.52 

1.62 

3 

4 

3.39 

2.12 

3.38 

2.13 

3.37 

2.15 

3.36 

2.16 

4 

6 

4.24 

2.65 

4.23 

2.67 

4.22 

2.69 

4.21 

2.70 

5 

6 

5.09 

3.18 

5.07 

3.20 

5.06 

3.22 

5.05 

3.25 

6 

7 

5.94 

3.71 

5.92 

3.74 

5.90 

3.76 

5.89 

3.79 

7 

■ 8 

6.78 

4-24 

6.77 

4.27 

6.75 

4.30 

6.73 

4.33 

8 

9 

7.63 

4.77 

7.61 

4.80 

7.59 

4.84 

7.57 

4.87 

9 

10 

8.48 

5.30 

8.46 

5.34 

8.43 

5.37 

8.41 

5.41 

10 

11 

9.33 

5.83 

9.30 

5.87 

9.28 

5.91 

9.25 

5.95 

11 

12 

10.18 

6.36 

10.15 

6.40 

10.12 

6.45 

10.09 

6.49 

12 

13 

11.02 

6.89 

10.99 

6.94 

10.96 

6.98 

10.93 

7.03 

13 

14 

11.87 

7.42 

11.84 

7.47 

11.81 

7.52 

11.77 

7.57 

14 

15 

12.72 

7.95 

12.69 

8.00 

12.65 

8.06 

12.62 

8.11 

15 

16 

13.57 

8.48 

13.53 

8.54 

13.49 

8.60 

13.46 

8.66 

16 

17 

14.42 

9.01 

14.38 

9.07 

14.34 

9.13 

14.30 

9.20 

17 

18 

16.26 

9.54 

15.22 

9.61 

15.18 

9.67 

15.14 

9.74 

18 

19 

16.11 

10.07 

16.07 

10.14 

16.02 

10.21 

15.98 

10,28 

19 

20 

16.96 

10.60 

16.91 

10.67 

16.87 

10.75 

16.82 

10,82 

20 

21 

17.81 

11.13 

17.76 

11.21 

17.71 

11.28 

17.66 

11.36 

21 

22 

18.66 

1 1 .66 

18.61 

11.74 

18.55 

11.82 

18.50 

11.90 

22 

23 

19.51 

12.19 

19.45 

12.27 

19.40 

12.36 

19.34 

12.44 

23 

24 

20.35 

12.72. 

20.30 

12.81 

20.24 

12.90 

20.18 

12.98 

24 

25 

21.20 

13.25 

21.14 

13.34 

21.08 

13.43 

21.03 

13.52 

25 

26 

22.05 

13.78 

21.99 

13.87 

21.93 

13.97 

21.87 

14.07 

26 

27 

22.90 

14.31 

22.83 

14.41 

22.77 

14.51 

22.71 

14.61 

27 

28 

23.75 

14.84 

23.68 

14.94 

23.61 

15.04 

23.55 

15.15 

28 

29 

24.59 

15.37 

24.53 

15.47 

24.46 

15.58 

24.39 

15.69 

29 

30 

25.44 

15.90 

25.37 

16.01 

25.30 

16.12 

25.23 

16.23 

30 

31 

26.29 

16.43 

26.22 

16.54 

26.15 

16.66 

26.07 

16.77 

31 

32 

27.14 

16.96 

27.06 

17.08 

26.99 

17.19 

26.91 

17.31 

32 

33 

27.99 

17.49 

27.91 

17.61 

27.83 

17.73 

27.75 

17.85 

33 

34 

28.83 

18.02 

28.75 

18.14 

28.68 

18.27 

28.60 

18.39 

34 

35 

29.68 

18.55 

29.60 

18.68 

29.52 

18.81 

29.44 

18.93 

35 

36 

30.53 

19.08 

30.45 

19.21 

30.36 

19.34 

30.28 

19.48 

36 

37 

31.38 

19.61 

31.29 

19.74 

31.21 

19.88 

31.12 

20.02 

37 

38 

32.23 

20.14 

32.14 

20.28 

32.05 

20.42 

31.96 

20.56 

38 

39 

33.07 

20.67 

32.98 

20.81 

32.89 

20.95 

32.80 

21.10 

39 

40 

33.92 

21.20 

33.83 

21.34 

33.74 

21.49 

33.64 

21 .64 

40 

41 

34.77 

21.73 

34.67 

21.88 

34.58 

22.03 

34.48 

22.18 

41 

42 

35.62 

22.26 

35.52 

22.41 

35.42 

22.57 

35.32 

22.72 

42 

43 

36.47 

22.79 

36.37 

22.95 

36.27 

23.10 

36.16 

23.26 

43 

44 

37.31 

23.32 

37.21 

23.48 

37.11 

23.64 

37.01 

23.80 

44 

45 

38.16 

23.85 

38.06 

24.01 

37.95 

24.18 

37.85 

24.34 

45 

46 

39.01 

24.38 

38.90 

24.55 

38.80 

24.72 

38.69 

24.88 

46 

47 

39.86 

24.91 

39.75 

25.08 

39  ..64 

25.25 

39.53 

25.43 

47 

48 

40.71 

25.44 

40.59 

25.61 

40.48 

25.79 

40.37 

25.97 

48 

49 

41.55 

25.97 

41.44 

26.15 

41.33 

26.33 

41.21 

26.51 

49 

50 

42.40 

26.50 

42.29 

26.68 

42.17 

26.86 

42.05 

27.05 

50 

8 

a 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dcp. 

Lat. 

O 

C 

1 

Q 

58  Dog. 

57$  Dog. 

57£  Deg. 

57$  Deg. 

d 

(A 

a 

TRAVERSE  TABLE, 


137 


o 

5T 

32  Deg. 

32*  Deg. 

32$  Deg. 

32f  Deg. 

a 

K" 

pi 

o 

a 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

o 

CD 

51 

43.25 

27*03 

43. 13 

27.21 

43.01 

27.40 

42.89 

27.59 

51 

52 

44.10 

27.56 

43.98 

27.75 

43.86 

27.94 

43.73 

28.13 

52 

53 

44.95 

28.09 

44.82 

28.28 

44.70 

28.48 

44.58 

28.67 

63 

54 

45.79 

28.62 

45.67 

28.82 

45.54 

29.01 

45.42 

29.21 

54 

55 

46.64 

29.15 

46.51 

29.35 

46.39 

29.55 

46.26 

29.75 

55 

56 

47.49 

29.68 

47.36 

29.88 

47.23 

30.09 

47.10 

30.29 

56 

57 

48.34 

30.21 

48.21 

30.42 

48.07 

30.63 

47.94 

30.84 

57 

58 

49.19 

30.74 

49.05 

30.95 

48.92 

31.16 

48.78' 

31.38 

58 

59 

50.03 

31.27 

49.90 

31.48 

49.76 

31.70 

49.62 

31.92 

59 

60 

50.88 

31.80 

50.74 

32.02 

50.60 

32.24 

50.46 

32.46 

60 

61 

51.73 

32.33 

51 .59 

32.55 

51.45 

32.78 

51.30 

33.00 

61 

62 

52.58 

32.85 

52.44 

33.08 

52.29 

33.31 

52.14 

33.54 

62 

63 

53.43 

33.38 

53.28 

33.62 

53.13 

33.85 

52.99 

34.08 

63 

64 

54.28 

33.91 

54.13 

34.15 

53.98 

34.39 

53.83 

34.62 

64 

65 

55.12 

34.44 

54.97 

34.68 

54.82 

34.92 

54.67 

35.16 

65 

66 

55 . 97 

34.97 

55.82 

35.22 

55.66 

35.40 

55.51 

35.70 

66 

67 

56.82 

35.50 

56.66 

35.75 

56.51 

36.00 

56.35 

36.25 

67 

68 

57.67 

36.03 

57.51 

36.29 

57.35 

36.54 

57. 19 

36.79 

68 

69 

58.52 

36.56 

58.36 

36.82 

58.19 

37.07 

58.03 

37.33 

69 

70 

59.36 

37.09 

59.20 

37.35 

59.04 

37.61 

58.87 

37.87 

70 

71 

60.21 

37.62 

60.05 

37.89 

59.88 

38.15 

59.71 

38.41 

71 

72 

61.06 

38.15 

60.89 

38.42 

60.72 

38.69 

60.55 

38.95 

72 

73 

61.91 

38.68 

61.74 

38.95 

61.57 

39.22 

61.40 

39.49 

73 

74 

62.76 

39.21 

62.58 

39.49 

62.41 

39.76 

62.24 

40.03 

74 

75 

63.60 

39.74 

63.43 

40.02 

63.25 

40.30 

63.08 

40.57 

75 

76 

64.45 

40.27 

64.28 

40.55 

64.10 

40.83 

63.92 

41.11 

76 

77 

65.30 

40.80 

65.12 

41.09 

64.94 

41.37 

64.76 

41.65 

77 

78 

66.15 

41.33 

65.97 

41.62 

65.78 

41.91 

65.60 

42.20 

78 

79 

67.00 

41.86 

66.81 

42.16- 

66.63 

42.45 

66.44 

42.74 

79 

80 

67.84 

42.39 

67.66 

42.69 

67.47 

42.98 

67.28 

43.28 

80 

81 

68.69 

42.92 

68.50 

43.22 

68.31 

43.52 

68.12 

43.82 

81 

82 

69.54 

43.45 

69.35 

43.76 

69.16 

44.06 

68.97 

44.36 

82 

83 

70.39 

43.98 

70.20 

44.29 

70.00 

44.60 

69.81 

44.90 

83 

84 

71*24 

44.51 

71.04 

44.82 

70.84 

45.13 

70.65 

45.44 

84 

85 

72.08 

45.04 

71 .89 

45.36 

71.69 

45.67 

71.49 

45.98 

85 

86 

72.93 

45.57 

72.73 

45.89 

72.53 

46.21 

72.33 

46.52 

86 

87 

73.78 

46.1.0 

73.58 

46.42 

73.38 

46.75 

73.  17 

47.06 

87 

88 

74.63 

46.63 

74.42 

46.96 

74.22 

47.28 

74.01 

47.61 

88 

89 

75.48 

47. 16 

75.27 

47.49 

75.06 

47.82 

74.85 

48.15 

89 

90 

76.32 

47.69 

76.12 

48.03 

75.91 

48.36 

75.69 

48.69 

90 

91 

77.17 

48.22 

76.96 

48.56 

76.75 

48.89 

76.53 

49.23 

91 

92 

78.02 

48.75 

77.81 

49.09 

77.59 

49.43 

77.38 

49  77 

92 

93 

78.87 

49.28 

78.65 

49.63 

78.44 

49.97 

78.22 

50.31 

93 

94 

79,72 

49.81 

79.50 

50.16 

79.28 

50.51 

79.06 

50.85 

94 

95 

80.56 

50.34 

80.34 

50.69 

80.12 

51.04 

79.90 

51.39 

95 

96 

81.41 

50.87 

81.19 

51.23 

80.97 

51.58 

80.74 

51.93 

96 

97 

82.26 

51.40 

82.04 

51.76 

81.81 

52. 12 

81.58 

52.47 

97 

98 

83.11 

51.93 

82.88 

52.29 

82.65 

52. 66 

82.42 

53.02 

98 

99 

83.96 

52.46 

83.73 

52.83 

83.50 

53.19 

83.26 

53.56 

99 

100 

84.80 

52.99 

84.57 

53.36 

84.34 

53.73 

84.10 

54.10 

100 

6 

CJ 

c 

Dep. 

Lai. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

o 

c 

ri 

VJ 

Q 

68  Deg. 

57|  Deg. 

57$  Deg. 

57*  Deg. 

09 

5 

138 


TRAVERSE  TABLE. 


c 

33  Deg. 

33*  Deg. 

33*  Deg 

33*  Deg. 

O 

£ 

3 

c: 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep.  ! 

? 

a 

1 

0.84 

0.54 

0.84 

0.55 

0.83 

0.55 

#83 

~0.fi6 

1 

2 

1.68 

1.09 

1.67 

1.10 

1.67 

l . 10 

1 .6G 

1.11 

2 

3 

2 . 52 

1.63 

2.51 

1.64 

2.50 

1.66 

2.49 

1 67  1 

3 

4 

3.35 

2.18 

3.35 

2.19 

3.34 

2 21 

3.33 

2.22  , 

4 

5 

4 19 

2 72 

4.18 

2.74 

4. 17 

2.76 

4. 16 

2.78  ; 

5 

0 

5 03 

3 27 

5.02 

3.29 

5.00 

3.31 

4.99 

3.33 

6 

7 

5.87 

3 81 

5.85 

3.84 

5.84 

3.86 

5.82 

3.89  | 

7 

8 

6.71 

4.36 

6.69 

4.39 

6.67 

4.42 

6.65 

4.44 

8 

9 

7.55 

4.90 

7.53 

4.93 

7.50 

4.97 

7.48 

5.00 

) 

10 

8.39 

5.45 

8.36 

5.48 

8.34 

5.52 

8.31 

5.56 

10 

11 

9.23 

5.99 

9.20 

6.03 

9.17 

6.07 

9.15 

6.11 

1 f 

12 

10.06 

6.54 

10.04 

6.58 

10.01 

6.62 

9.98  | 

6.67 

12 

13 

10.90 

7.08 

10.87 

7. 13 

10.84 

7.18 

10.81 

7.22 

13 

14 

11.74 

7.62 

11.71 

7.68 

11.67 

7.73 

11.64 

7.78 

14 

15 

12.58 

8.17 

12.54 

8.22 

12.51 

8.28 

12.47 

8.33 

15 

16 

13.42 

8.71 

13.38 

8.77 

13.34 

8.83 

13.30 

8.89 

16 

17 

14.26 

9.26 

14.22 

9.32 

14.18 

9.38 

14.13 

9.44 

17 

18 

15. 10 

9.80 

15.05 

9.87 

15.01 

9.93 

14.97 

10.00 

18 

19 

15.93 

10.35 

15.89 

10.42 

15.84 

10.49 

15.80 

10.56 

19 

20 

16.77 

10.89 

16.73 

10.97 

16.68 

11.04 

16.63 

11.11 

20 

21 

17.61 

11.44 

17.56 

11.51 

17.51 

11.59 

17.46 

11.67 

21 

22 

18.45 

11.98 

18.40 

12.06 

18.35 

12.14 

18.29 

12.22 

22 

23 

19.29 

12.53 

19.23 

12.61 

19.18 

12.69 

19.12 

12.78 

23 

24 

20.13 

13.07 

20.07 

13.16 

20.01 

13.25 

19.96 

13.33 

24 

25 

20.97 

13.62 

20.91 

13.71 

20.85 

13.80 

20.79 

13.89 

25 

26 

21.81 

14.16 

21.74 

14.26 

21.68 

14.35 

21.62 

14.44 

26 

27 

22.64 

14.71 

22.58 

14.80 

22.51 

14.90 

22.45 

15.00 

27 

28 

23.48 

15.25 

23.42 

15.35 

23.35 

15.45 

23.28 

15.56 

23 

29 

24.32 

15.79 

24.25 

15.90 

24.18 

16.01 

24.11 

16.11 

29 

30 

25.16 

16.34 

25.09 

! 16.45 

25.02 

16.56 

24.94 

16.67 

30 

31 

26.00 

16.88 

25.92 

17.00 

25.85 

17.11 

25.78 

17.22 

'31 

32 

26.84 

17.43 

26.76 

17.55 

26.68 

17.66 

26.61 

17.78 

32 

33 

27.68 

17.97 

27.60 

18.09 

27.52 

18.21 

27.44 

18.33 

33 

34 

28.51 

18.52 

28.43 

18.64 

28.35 

18.77 

28.27 

18.89 

34 

35 

29.35 

19.06 

29.27 

19.19 

29.19 

19.32 

29.10 

19.44 

35 

36 

30.19 

19.61 

30.11 

19.74 

30.02 

19.87 

29.93 

20.00 

36 

37 

31.03 

20.15 

30.94 

20.29 

30.85 

20.42 

30.76 

20.56 

37 

38 

31.87 

20.70 

31.78 

20.84 

31.69 

20.97 

31.60 

21.11 

38 

39 

32  71 

21.24 

32.62 

21.38 

32.52 

21.53 

| 32.43 

21.67 

39 

40 

33.55 

21.79 

33.45 

21.93 

33.36 

22.  G8 

33.26 

22.22 

40 

41 

34.39 

22.33 

34.29 

22.48 

34.19 

22.63 

34.09 

22.78 

41 

42 

35.22 

22.87 

35.12 

23 . 03 

35 . 02 

23.18 

34.92 

23.33 

12 

43 

1 36.06 

23.42 

35.96 

23.58 

35.86 

23.73 

35.75 

23.89 

43 

44 

1 36.90 

23.96 

36.80 

24. 12 

36 . 69 

24.29 

36.58 

24.45 

44 

45 

37.74 

24.51 

37.63 

24.67 

37.52 

24.84 

37.42 

25  00 

45 

46 

38 . 58 

25 . 05 

38.47 

25.22 

38.36 

25.39 

38.25 

25.56 

46 

47 

39.42 

25 . 60 

39.31 

25.77 

39.19 

25.94 

39.08 

26.11 

47 

18 

40.26 

26.14 

40.14 

26.32 

40.03 

26.49  1 

39.91 

26.67 

i 48 

49 

41.09 

26.69 

40.98 

26.87 

40.86 

27.04 

40.74 

27.22 

49 

50 

41.93 

27.23 

41.81 

27.41 

41.69 

27.60 

41.57 

27.78 

| 50 

S 

3 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  j Lat. 

Dep. 

Lat. 

! ® 

1 1 

3 

.a 

Q 

57  Deg. 

56*  Deg. 

56*  Deg. 

56*  Dog. 

3 

1 

traverse  table. 


130 


o 

s 

33  Dog. 

33*  Deg. 

33j  Deg. 

33|  Deg.  1 

Distance,  j 

3 

O 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

61 

42.77 

27.78 

42.65 

27.96 

42753 

28715 

42740" 

28.33 

51 

52 

43.61 

28.32 

43.49 

28.51 

43.36 

28 . 70 

43.24 

28.89 

52 

53 

44.45 

28.87 

44.32 

29.06 

44.20 

29.25 

44.07 

29.45 

53 

64 

45.29 

29.41 

45.16 

29.61 

45.03 

29.80 

44.90 

30.00 

54 

66 

46.13 

29.96 

46.00 

30.16 

45.86 

30.36 

45.73 

30.56 

55 

66 

46.97 

30.50 

46.83 

30.70 

46.70 

30.91 

46.56 

31.11 

56 

67 

47.80 

31.04 

47.67 

31.25 

47.53 

31.46 

47.39 

31.67 

57 

68 

48.64 

31.59 

48.50 

31.80 

48.37 

32.01 

48.23 

32.22 

58 

59 

49.48 

32.13 

49.34 

32.35 

49.20 

32.56  1 

49.06  ! 

32  78 

59 

60 

50.32 

32.68 

50.18 

32.90 

50.03 

33.12  i 

49.89 

33.33 

60 

61 

51.16 

33.22 

51 .01 

33.45 

50.87 

33.67 

50.72 

33.89 

61 

62 

52.00 

33.77  1 

51.85 

33.99 

51.70 

34.22 

51.55 

34.45 

62 

63 

52.84 

34.31 

52.69 

34.54 

52.53 

34.77 

52.38 

35.00 

63 

64 

53.67 

34.86 

53.52 

35.09 

53.37 

35.32 

53.21 

35.56 

64 

65 

54.51 

35.40 

54.36 

35.64 

54.20 

35.88 

54.05 

36.11 

65 

66 

55.35 

35.95 

55.19 

36.19 

55.04 

36.43 

54.88 

36.67 

PJ 

67 

56.19 

36.49 

56.03 

36.74 

55.87 

36.98 

55.71 

37.22 

67 

68 

57.03 

37.04 

56.87 

37.28 

56.70 

37.53 

56.54 

37.78 

68 

69 

57.87 

37.58 

57.70 

37.83 

57.54 

38.08 

57.37 

38.33 

69 

70 

58.71 

38.12 

58.54 

38.38 

58.37 

38.64 

58.20 

38.89. 

70 

71 

59.55 

38.67 

59.38 

38.93 

59.21 

39.19 

59.03 

39.45 

71 

72 

60.38 

39.21 

60.21 

39.48 

60.04 

39.74 

59.87 

40.00 

72 

73 

61.22 

39.76 

61.05 

40.03 

60.87 

40.29 

60.70 

40.56 

73 

74 

62.06 

40.30 

61.89 

40.57 

61.71 

40.84 

61.53 

41.11 

74 

75 

62.90 

40.85 

62.72 

41.12 

62.54 

41.40 

62.36 

41.67 

75 

76 

63.74 

41.39 

63.56 

41.67 

63.38 

41.95 

63.19 

42.22 

76 

77 

64.58 

41.94 

64.39 

42.22 

64.21 

42.50 

64.02 

42.78 

77 

78 

65.42 

42.48 

65.23 

42.77 

65.04 

43.05 

64.85 

43.33 

78 

79 

06.25 

43.03 

66.07 

43.32 

65.88 

43.60 

65.69 

43.89 

79 

80 

67.09 

43.57 

66.90 

43.86 

66.71 

44.15 

66.52 

44.45 

80 

81 

67.93 

44.12 

67.74 

44.41 

67.54 

44.71 

67.35 

45.00 

81 

82 

68.77 

44.66 

68.58 

44.96 

68.38 

45.26 

68.18 

45.56 

82 

83 

69.61 

45.20 

69.41 

45.51 

69.21 

45.81 

69.01 

46.11 

83 

84 

70.45 

45.75 

70 . 25 

46.06 

70.05 

46.36 

69.84 

46.67 

84 

85 

71.29 

46.29 

71.08 

46.60 

70.88 

46.91 

70.67 

47.22 

85 

86 

72.13 

46.84 

71.92 

47.15 

71.71 

47.47 

71.51 

47.78 

86 

87 

72.96 

47.38 

72.76 

47.70 

72.55 

48.02 

72.34 

48.33 

87 

88 

73.80 

47.93 

73.59 

48.25 

73.38 

48.57 

73.17 

48.89 

88 

89 

74.64 

48.47 

74.43 

48.80 

74.22 

49.12 

74.00 

49.45 

89 

90 

75.48 

49.02 

75.27 

49.35 

75.05 

49.67 

74.83 

50.00 

90 

91 

70.32 

49.56 

76.10 

49.89 

75.88 

50.23 

75.66 

50.56 

91 

92 

77.16 

50.11 

76.94 

50.44 

76.72 

50.78 

76.50 

51.11 

92 

93 

78.00 

50.65 

77.77 

50.99 

77.55 

51.33 

77.33 

51.67 

, 93 

94 

78  83 

61.20 

78.61 

51.54 

78.39 

51.88 

78.16 

52.22 

94 

95 

79.67 

51.74 

79.45 

52.09 

79.22 

52  43 

78.99 

52.78 

95 

96 

80.51 

62.29  1 

1 80.28 

52.64 

80.05 

52.99 

79.82 

53.33 

96 

97 

81 .35 

52.83 

81.12 

53.18 

80.89 

53.54 

1 80.65 

53.89 

97 

98 

82.19 

53.37 

81.96 

53.73 

81.72 

54.09 

81.48 

54.45 

98 

99 

83.03 

63.92 

82.79 

54.28 

82.55 

54.64 

82.32 

55.00 

99 

100 

83.87 

1 54.46 

83.63 

54.83 

83.39 

j 55.19 

83.15 

55.56 

00 

l 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| Lat. 

Dep. 

Lat. 

I 8 
§ 

!i 

Q 

j 57  Deg. 

565  Deg. 

5 6* 

Deg. 

50*  Dog. 

1 

140 


TRAVERSE  TABLE. 


Distance. 

34  Deg. 

34^  Deg. 

34£ 

Deg. 

34|  Deg. 

0 

1 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.83 

0.56 

0.83 

0.56 

0.82 

0.57 

0.  82 

0.57 

1 

2 

1.66 

1.12 

1.65 

1.13 

1.65 

1.13 

1.64 

1.14 

2 

3 

2.49 

1.68 

2.48 

1.69 

2.47 

1.70 

2.46 

1.71 

3 

4 

3.32 

2.24 

3.31 

2.25 

3.30 

2.27 

3.29 

2.28 

4 

5 

4.15 

2.80 

4.13 

2.81 

4.12 

2.83 

4.11 

2.85 

5 

6 

4.97 

3.36 

4.96 

3.38 

4.94 

3.40 

4.93 

3.42 

6 

7 

5.80 

3.91 

5.79 

3.94 

5.77 

3.9G 

5.75 

3.99 

7 

8 

6.63 

4-47 

6.61 

4.50 

6.59 

4.53 

6.57 

4.56 

8 

9 

7.46 

5.03 

7.44 

5.07 

7.42 

5.10 

7.39 

5.13 

9 

10 

8.29 

5.59 

8.27 

5.63 

8.24 

5.66 

8.22 

5.70 

10 

11 

9.12 

6.15 

9.09 

6.19 

9.07 

6.23 

9.04 

6.27 

11 

12 

9.95 

6.71 

9.92 

6.75 

9.89 

6.80 

9.86 

6.84 

12 

13 

10.78 

7.27 

10.75 

7.32 

10.71 

7.36 

10.68 

7.41 

13 

14 

11.61 

7.83 

11.57 

7.88 

11.54 

7.93 

11.50 

7.98 

14 

15 

12.44 

8.39 

12.40 

8.44 

12.36 

8.50 

12.32 

8.55 

15 

16 

13.26 

8.95 

13.23 

9.00 

13.19 

9.06 

13.15 

9.12 

16 

17 

14.09 

9.51 

14.05 

9.57 

14.01 

9.63 

13.97 

9.69 

17 

18 

14.92 

10.07 

14.88 

10.13 

14.83 

10.20 

14.79 

10.26 

18 

19 

15.75 

10.62 

15.71 

10.69 

15.66 

10.76 

15.61 

10.83 

19 

20 

16.58 

11.18 

16.53 

11.26 

16.48 

11.33 

16.43 

11.40 

20 

21 

17.41 

11.74 

17.36 

11.82 

17.31 

11.89 

17.25 

11.97 

21 

22 

18.24 

12.30 

18.18 

12.38 

18.13 

12.46 

18.08 

12.54 

22 

23 

19.07 

12.86 

19.01 

12.94 

18.95 

13.03 

18.90 

13.11 

23 

24 

19.90 

13.42 

19.84 

13.51 

19.78 

13.59 

19.72 

13.68 

24 

25 

20.73 

13.98 

20.66 

14.07 

20.60 

14.16 

20.54 

14.25 

25 

26 

21.55 

14.54 

21.49 

14.63 

21.43 

14.73 

21 .36 

14.82 

26 

27 

22.38 

15.10 

22.32 

15.20 

22.25 

15.29 

22. 18 

15.39 

27. 

28 

23.21 

15  66 

23.14 

15.76 

23.08 

15.86 

23.01 

15.96 

28 

29 

24.04 

16.22 

23.97 

16.32 

23.90 

16.43 

23.83 

16.53 

29 

30 

24.87 

16.78 

24.80 

16.88 

24.72 

16.99 

24.65 

17.10 

30 

31 

25.70 

17.33 

25.62 

17.45 

25.55 

17.56 

25.47 

17.67 

31 

32 

26.53 

17.89 

26.45 

18.01 

26.37 

18.12 

26.29 

18.24 

32 

33 

27.36 

18.45 

27.28 

18.57 

27.20 

18.69 

27.11 

18.81 

33 

34 

28.19 

19.01 

28.10 

19.14 

28.02 

19.26 

27.94 

19.38 

34 

35 

29.02 

19.57 

28.93 

19.70 

28.84 

19.82 

28.76 

19.95 

35 

36 

29.85 

20.13 

29.76 

20.26 

29.67 

20.39 

29.58 

20.52 

36 

37 

30.67 

20.69 

30.58 

20.82 

30.49 

20.96 

30.40 

21.09 

37 

38 

31.50 

21.25 

31.41 

21.39 

31.32 

21.52 

31.22 

21.66 

38 

39 

32.33 

21.81 

32.24 

21.95 

32. 14 

22.09 

32.04 

22.23 

39 

40 

33.16 

22.37 

33.06 

22.51 

32.97 

22.66 

32.87 

22.80 

40 

41 

33.99 

22.93 

33.89 

23.07 

33.79 

23.22 

33.69 

23.37 

41 

42 

34.82 

23.49 

34.72 

23.64 

34.61 

23.79 

34.51 

23.94 

42 

43 

35.65 

24.05 

35.54 

24.20 

35.44 

24.36 

35.33 

24.51 

43 

44 

36.48 

24.60 

36.37 

24.76 

36.26 

24.92 

36.15 

25.08 

44 

45 

37.31 

25.16 

37.20 

25.33 

37.09 

25.49 

36.97 

25.65 

45 

46 

38.14 

25.72 

38.02 

25.89 

37.91 

26.05 

37.80 

26.22 

46 

47 

38.96 

26.28 

33.85 

26.45 

38.73 

26.62 

3S-62 

26.79 

47 

48 

39.79 

26.84 

39.68 

27.01 

39.56 

27.19 

39.44 

27.36 

48 

49 

40.62 

27.40 

40.50 

27.58 

40.38 

27.75 

40.26 

27.93 

49 

50 

41.45 

27.96 

41.33 

28.14 

41.21 

28.32 

41.08 

28.50 

50 

< D 

O 

£ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

o 

c 

to 

S 

56  Deg. 

55J  Deg. 

n 55%  Deg. 

55-J  Deg. 

d 

V) 

s 

TRAVERSE  TABLE, 


141 


o 

55 

34  Deg. 

34$  Deg. 

341 

Deg. 

34$  Deg. 

C 

CO* 

pr 

s 

o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D 

o 

© 

51 

42.28 

28.52 

42.16 

28.70 

42.03 

28.89 

41.90 

29.07 

~5l 

52 

43.11 

29.08 

42.98 

29.27 

42.85 

29.45 

42.73 

29 . 64 

52 

53 

43.94 

29.64 

43.81 

29.83 

43.68 

30.02 

43.55 

30.21 

53 

54 

44.77 

30.20 

44.64 

30.39 

44.50 

30.59 

44.37 

30.78 

54 

55 

45.60 

30.76 

45.46 

30.95 

45.33 

31.15 

45.19 

31.35 

55 

56 

46.43 

31.31 

46.29 

31.52 

46.15 

31.72 

46.01 

31.92 

56 

57 

47.26 

31.87 

47.12 

32.08 

46.98 

32.29 

46.83 

32.49 

57 

58 

48.08 

32.43 

47.94 

32.64 

47.80 

32.85 

47.66 

33.06 

58 

59 

48.91 

32.99 

48.77 

33.21 

48.62 

33.42 

48.48 

33.63 

59 

60 

49.74 

33.55 

49.60 

33.77 

49.45 

33.98 

49.30 

34.20 

60 

61 

50.57 

34.11 

50.42 

34.33 

50.27 

34.55 

50.12 

34.77 

61 

62 

51.40 

34.67 

51.26 

34.89 

51.10 

35.12 

50.94 

35.34 

62 

63 

52.23 

35.23 

52.08 

35.46 

51.92 

35.68 

51.76 

35.91 

63 

64 

53.06 

35.79 

52 . 90 

36.02 

52.74 

36.25 

52.59 

36.48 

64 

65 

53.89 

36.35 

53.73 

36.58 

53.57 

36.82 

53.41 

37.05 

65 

66 

54.72 

36.91 

54.55 

37.15 

54.39 

37,38 

54.23 

37.62 

66 

67 

55.55 

37.46 

55.38 

37.71 

55.22 

37.95 

55.05 

38.19 

67 

68 

50.37 

38.03 

56.21 

38.27 

56.04 

38.52 

55.87 

38.76 

68 

69 

57.20 

38.58 

57.03 

38.83 

56.86 

39.08 

56.69 

39.33 

69 

70 

58.03 

39.14 

57.86 

39.40 

57.69 

39.65 

57.52 

39.90 

70 

71 

58.86 

39.70 

58.69 

39,96 

58.51 

40.21 

58.34 

40.47 

71 

72 

59.69 

40.26 

59.51 

40.52 

59.34 

40.78 

59.16 

41 .04 

72 

73 

60.52 

40.82 

60.34 

41.08 

60.16 

41.35 

59.98 

41.61 

73 

74 

61.35 

41.38 

61.17 

41.65 

60.99 

41.91 

60.80 

42.18 

74 

75 

62.18 

41.94 

61.99 

42.21 

61.81 

42.48 

61.62 

42.75 

75 

76 

63.01 

42.50 

62.82 

42.77 

62.63 

43.05 

62.45 

43.32 

76 

77 

63.84 

43.06 

63.65 

43.34 

63.46 

43.61 

63.27 

43.89 

77 

78 

64.66 

43.62 

64.47 

43.90 

64.28 

44.18 

64.09 

44.46 

78 

79 

65.49 

44.18 

65.30 

44.46 

65.11 

44.75 

64.91 

45.03 

79 

80 

66.32 

44.74 

66.13 

45.02 

65.93 

45.31 

65.73 

45.60 

80 

81 

67.15 

45.29 

66.95 

45.59 

66.75 

45.88 

66  j55 

46.17 

81 

82 

67.98 

'45.85 

67.78 

46.15 

67.58 

46.45 

67.37 

46.74 

82 

83 

68.81 

46.41 

68.61 

46.71 

68.40 

47.01 

68.20 

47.31 

83 

84 

69. 64 

46.97 

69.43 

47.28 

69.23 

47.58 

69.02 

47.88 

84 

85 

70.47 

47.53 

70.26 

47.84 

70.05 

48.14 

69.84 

48.45 

85 

86 

71.30 

48.09 

71.09 

48.40 

70.87 

48.71 

70.66 

49.02 

86 

87 

72.13 

48.65 

71.91 

48.96 

71.70 

49.28 

71.48 

49.59 

87 

88 

72.96 

49.21 

72.74 

49.53 

72.52 

49.84 

72.30 

50. 16 

88 

89 

73.78 

49.77 

73.57 

50.09 

73.35 

50.41 

73.13 

50.73 

89 

90 

74.61 

50.33 

74.39 

50 . 65 

74.17 

50.98 

73.95 

51.30 

90 

91 

75.44 

50.89 

75.22 

51.22 

75.00 

51.54 

74.77 

51.87 

91 

92 

76.27 

51.45 

76.05 

51.78 

75.82 

52.11 

75 . 59 

52.44 

92 

93 

77.10 

52.00 

76.87 

52.34 

76.64 

52.68 

76.41 

53.01 

93 

94 

77.93 

52.56 

77.70 

52.90 

77.47 

53.24 

77.23 

53.58 

94 

95 

78.76 

53.12 

78.53 

53.47 

78.29 

53.81 

78.06 

54.15 

95 

96 

79.59 

53.68 

79.35 

54.03 

79.12 

54.37 

78.88 

54.72 

96 

97 

80.42 

54.24 

80.18 

54.59 

79.94 

54.94 

79.70 

55.29 

97 

98 

81.25 

54.80 

81.01 

55.15 

80.76 

55.51 

80.52 

55.86 

98 

99 

82.07 

55.36 

81.83 

55.72 

81.59 

56.07 

81.34 

56.43 

99 

100 

82.90 

55.92 

82.66 

56.28 

82.41 

56.64 

82.16 

57.00 

100 

© 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

s 

C 

03 

5 

56  Deg. 

55|  Deg. 

554 

Deg. 

55$  Deg. 

Q 

142 


TRAVERSE  TABLE, 


g 

C »* 

35  Deg. 

354  Deg. 

35$  Dog. 

35f  Deg. 

O 

</>* 

b 

8 

L<it« 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

S 

8 

1 

0.82 

0.57 

0.82 

0.58 

0.81 

0.58 

0.81 

0.58 

i 

2 

1.64 

1.15 

1.63 

1.15 

1.63 

1.16 

1.62 

1.17 

2 

3 

2.46 

1.72 

2.45 

1.73 

2.44 

1.74 

2.43 

1.75 

3 

4 

3.28 

2.29 

3.27 

2.31 

3.26 

2.32 

3.25 

2.34 

4 

5 

4.  10 

2.87 

4.08 

2.89 

4.07 

2.90 

4.06 

2.92 

rj 

6 

4.91 

3.44 

4.90 

3.46 

4.88 

3.48 

4.87 

3.51 

0 

7 

5.73 

4.01 

5.72 

4.04 

5.70 

4.06 

5.68 

4.09 

7 

8 

6.55 

4.59 

6.53 

4.62 

6.51 

4.65 

6.49 

4.67 

8 

9 

7.37 

5.16 

7.35 

5.19 

7.33 

5.23 

7.30 

5.26 

9 

10 

8.19 

5.74 

8.17 

5.77 

8.14 

5.81 

8.12 

5.84 

10 

11 

9.01 

0.31 

8.98 

6.35 

8.96 

6.39 

8.93 

6.43 

11 

12 

9.83 

6.88 

9.80 

6.93 

9.77 

6.97 

9.74 

7.01 

12 

13 

10.65 

7.46 

10.62 

7.50 

10.58 

7.55 

10.55 

7.60 

13 

14 

11.47 

8.03 

11.43 

8.08 

11.40 

8.13 

11.36 

8.18 

14 

15 

12.29 

8.60 

12.25 

8.66 

12.21 

8.71 

12. 17 

8.76 

15 

16 

13.11 

9.18 

13.07 

9'.  23 

13.03 

9.29 

12.99 

9.35 

16 

17 

l3.93 

9.75 

13.88 

9.81 

13.84 

9.87 

13.80 

9.93 

17 

18 

14.74 

10.32 

14.70 

10.39 

14.65 

10.45 

14.61 

10.52 

18 

19 

15.56 

10.90 

15.52 

10.97 

15.47 

11.03 

15.42 

11.10 

19 

20 

16.38 

11.47 

16.33 

11.54 

16.28 

11.61 

16.23 

11.68 

20 

21 

17.20 

12.05 

17. 15 

12.12 

17. 10 

12.19 

17.04 

12.27 

21 

22 

18.02 

12.62 

17.97 

12.70 

17.01 

12.78 

17.85 

12.85 

22 

23 

18.84 

13. 19 

18.78 

13.27 

18.72 

13.36 

18.67 

13.44 

23 

24 

19.66 

13.77 

19.60 

13.85 

19.54 

13.94 

19.48 

14.02 

24 

25 

20.48 

14.34 

20.42 

14.43 

20.35 

14.52 

20.29 

14.61 

25 

26 

21.30 

14.91 

21.23 

15.0' 

21.17 

15.10 

21.10 

15.19 

26 

27 

22.12 

15.49 

22.05 

15.58 

21.98 

15.68 

21.91 

15.77 

27 

28 

22.94 

16.06 

22.87 

16.16 

22.80 

16.26 

22.72 

16.36 

28 

29 

23.76 

16.63 

23.68 

16.74 

23.61 

16.84 

23 . 54 

16.94 

29 

30 

24.57 

17.21 

24.50 

17.31 

24.42 

17.42 

24.35 

17.53 

30 

31 

25.39 

17.78 

25.32 

17.89 

25.24 

18.00 

25. 16 

18.11 

31 

32 

26.21 

18.35 

26.13 

18.47 

26.05 

18.58 

25.97 

18.70 

32 

33 

27.03 

18.93 

26.95 

19.05 

26.87 

19. 16 

26.78 

19.28 

33 

34 

27.85 

19.50 

27.77 

19.62 

27.68 

19.74 

27.59 

19.86 

34 

35 

28.67 

20.08 

28.58 

20.20 

28.49 

20.32 

28.41 

20.45 

35 

36 

29.49 

20.65 

29.40 

20.78 

29.31 

20.91 

29.22 

21.03 

36 

37 

30.31 

21.22 

30.22 

21.35 

30.12 

21.49 

30.03 

21,62 

37 

38 

31.13 

2 1 ! 80 

31.03 

21.93 

30.94 

22.07 

30.84 

22.20 

38 

39 

31.95 

22.37 

31.85 

22.51 

31.75 

22.65 

31.65 

22.79 

39 

40 

32 . 77 

22 . 94 

32 . 67 

23.09 

32.56 

23.23 

32.46 

23.37 

40 

41 

33.59 

22.52 

33.48 

23.66 

33.38 

23.81 

33.27 

23.95 

41 

42 

34.40 

24.09 

34 . 30 

24.24 

34.19 

24.39 

34.09 

24.54 

42 

43 

35.22 

24.66 

35.12 

24.82 

35.01 

21.97 

34.90 

25.12 

43 

44 

36.04 

25 . 24 

35.93 

25.39 

35.82 

25.55 

35.71 

25.71 

44 

45 

36.86 

25.81 

36.75 

25.97 

36.64 

26.13 

36.52 

26.29 

45 

46 

37.68 

26.38 

37.57 

26.55 

37.45 

26.71 

37.33 

26.88 

46 

47 

38 . 50 

26 . 96 

38.38 

27.13 

38.26 

27.29 

38.14 

27.46 

47 

48 

39.32 

27.53 

39.20 

27.70 

39.08 

27.87 

38.96 

28.04 

48 

49 

40. 14 

28.11 

40.02 

28.28 

39.89 

28.45 

39.77 

28.63 

49 

50 

40 . 96 

28 . 68 

40.83 

28.86 

40.71 

29.04 

40.58 

29.21 

50 

a> 

o 

q 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

<u 

o 

c 

3 

t/> 

s 

55  Deg. 

54|  Deg. 

54$  Deg. 

54*  Deg. 

3 

s 

TRAVERSE  TABLE, 


143 


D 

35  Deg. 

35$  Deg. 

35$  Deg, 

35 1 Deg. 

a 

po 

P 

O 

cd 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat, 

Dep. 

p 

O 

CD 

51 

41.78 

29.25 

41.65 

29.43 

41.52 

29.62 

41.39 

29.80 

"57 

52 

42.60 

29.83 

42.47 

30.01 

42.33 

30.20 

42.20 

30.38 

52 

53 

43.42 

.30.40 

43.28 

30.59 

43.15 

30.78 

43.01 

30.97 

53 

54 

44.23 

30.97 

44.10 

31.17 

43.96 

31.36 

43.82 

31  .55 

54 

55 

45.05 

31.55 

44.92 

31.74 

44.78 

31.94 

44.64 

32.  13 

55 

56 

45.87 

32.12 

45.73 

32.32 

45.59 

32.52 

45.45 

32 . 72 

56 

57 

46.69 

32.69 

46.55 

32.90 

46.40 

33.10 

46.26 

33.30 

57 

58 

47.51 

33.27 

47.37 

33.47 

47.22 

33.68 

47.07 

33.89 

58 

59 

48 . 33 

33.84 

48.18 

34.05 

48.03 

34.26 

47.88 

34.47 

59 

60 

49.15 

34.41 

49.00 

34.63 

48.85 

34.84 

48.69 

35 . 05 

60 

61 

49.97 

34.99 

49.82 

35.21 

49.66 

35.42 

49.51 

35.64 

61 

62 

50.79 

35 . 56 

50.63 

35.78 

50.48 

36.00 

50.32 

36 . 22 

62 

63 

51  .61 

36.14 

51.45 

36.36 

51.29 

36.58 

51.13 

36.81 

63 

64 

52.43 

36.71 

52.27 

36.94 

52.10 

37.16 

51.94 

37.39 

64 

65 

53.24 

37.28 

53.08 

37.51 

52.92 

37.75 

52.75 

37.98 

65 

66 

54.06 

37.86 

53 . 90 

38.09 

53.73 

38.33 

53.56 

38 . 56 

66 

67 

54.88 

38.43 

54.71 

38.67 

54.55 

38.91 

54.38 

39.14 

67 

68 

55 . 70 

39.00 

55.53 

39.25 

55.36 

39.49 

"55.19 

39.73 

68 

69 

56.52 

39.58 

58.35 

39.82 

56 . 1 7 

40.07 

56.00 

40.31 

69 

70 

57.34 

40.15 

57. 16 

40  40 

56.99 

40.65 

56.81 

40.90 

70 

71 

58.  16 

40.72 

57.98 

40.98 

57.80 

41 .23 

57.62 

41.48 

71 

72 

58.98 

41.30 

58.80 

41.55 

58 . 62 

41.81 

58.43 

42.07 

72 

73 

59.80 

41.87 

59.61 

42.13 

59 . 43 

42.39 

59.24 

42.65 

73 

74 

60.62 

42.44 

60.43 

42.71 

60.24 

42.97 

60.06 

43.23 

74 

75 

61 .44 

43.02 

61 .25 

43.29 

61.06 

43.55 

60.87 

43.82 

75 

76 

62.26 

43.59 

62.06 

43.86 

61.87 

44.13 

61.68 

44.40 

76 

77 

63.07 

44.17 

62.88 

44.44 

62.69 

44.71 

62.49 

44.99 

77 

78 

63.89 

44.74 

63.70 

45.02 

63.50 

45.29 

63.30 

45.57 

78 

79 

64.71 

45.31 

64.51 

45.59 

64.32 

45. 8S 

64.11 

46.16 

79 

80 

65.53 

45.89 

65.33 

46.17 

65.  13 

46.46 

64.93 

46.74 

80 

81 

66.35 

46.46 

66. 15 

46.75 

65.94 

47.04 

65.74 

47.32 

81 

82 

67.17 

47.03 

66.96 

47.33 

66 . 76 

47.62 

66.55 

47.91 

82 

83 

67.99 

47.61 

67.78 

47.90 

67.57 

48.20 

67.36 

48.49 

83 

84 

68.81 

48.18 

68.60 

48.48 

68.39 

48.78 

68.17 

48.08 

84 

85 

C9.63 

48.75 

69.41 

49.06 

69.20 

49.36 

68.98 

49.66 

85 

86 

70.45 

49.33 

70.23 

49.63 

70.01 

49.94 

69.80 

50.25 

86 

87 

71.27 

49.90 

71.05 

50.21 

70.83 

50.52 

70.61 

50.83 

87 

88 

72.09 

50  47 

71 .86 

50.79 

71.64 

51.10 

71.42 

51 .41 

88 

89 

72.90 

51.05 

72.68 

51.37 

72.46 

51.68 

72.23 

52 . 00 

89 

90 

73.72 

51.62 

73.50 

51.94 

73.2  T 

,52.26 

73.04 

52.58 

90 

91 

74.54 

52.20 

74.31 

52.52 

74.08 

52.84 

73.85 

53.17 

9L 

92 

75.36 

52.77 

75.13 

53.10 

74.90 

53.42 

74.66 

53.75 

92 

93 

76.18 

53.34 

75.95 

53.67 

75.71 

54.01 

75.48 

54.34 

93 

94 

77.00 

53.92 

76.76 

54.25 

76.53 

'54.59 

76.29 

54.92 

94 

95 

77.82 

54.49 

77.58 

54.83 

77.34 

55.17 

77.10 

55.50 

95 

96 

78.64 

55.08 

78.40 

55.41 

78.16 

55.75 

77.91 

56.09 

96 

97 

79.46 

55.84 

79.21 

55.98 

78  97 

56.33 

78.72 

56.67 

97 

98 

80.28 

56.21 

80.03 

56 . 56 

70.78 

56.91 

79.53 

57.26 

98 

99 

81.10 

56.78 

80.85 

57.14 

SO.  60 

57.49 

80.35 

57.84 

99 

J00 

81.92 

57.36 

81,66 

67.71 

SI  .41 

58.07 

81.16 

58.42 

100 

© 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

c 

cJ 

co 

Q 

66  Deg. 

54$  Deg. 

54$  Deg. 

54$  Degy, 

d 

m 

s 

144 


TRAVERSE  TABLE. 


| Distance.! 

i 

36  Deg. 

36*  Deg. 

36  j 

Dog. 

36*  Def 

5* 

ep. 

O 

m 

P 

a 

r> 

? 

j Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

1 

! 0.81 

0.69 

0.81 

0.59 

0.80 

0.59 

0.80' 

1 

.60 

i 

2 

1 1 .62 

1.18 

1.61 

1.18 

1.61 

1.19 

1.60 

1 

20 

* 

3 

1 2.43 

1.76 

2.42 

1.77 

2.41 

1.78 

2.40 

1 

.79 

3 

4 

3.24 

2.35 

3.23 

2.37 

3.22 

2.38 

3.20 

2 

.39 

4 

5 

4.05 

2.94 

4.03 

2.96 

4.02 

2.97 

4.01 

i 2 

.99 

5 

6 

4.85 

3.53 

4.84 

3.55 

4.82 

3.57 

4.81 

1 3 

.59 

6 

7 

5.66 

4.11 

5.65 

4.14 

5.63 

4.16 

5.61 

! 4 

.19 

7 

8 

6 47 

4.70 

6.45 

4.73 

6.43 

4.76 

6.41 

4 

.79 

8 

9 

7.28 

5.29 

7.26 

5.32 

7.23 

5.35 

7.21 

5 

.38 

9 

10 

! 8.09 

5.88 

8.06 

5 91 

8.04 

5.95 

8.01 

5 

.98 

10 

11 

| 8.90 

6.47 

8.87 

6.50 

8.84 

6.54 

8.81 

6 

.58 

'll 

12 

! 9.71 

7.05 

9.68 

7.10 

9.65 

7.14 

9.61 

7 

. 18 

12 

13 

10.52 

7.64 

10.48 

7.69 

10.45 

7.73 

10.42 

7 

.78 

13 

14 

11.33 

8.23 

11.29 

8.28 

11.25 

8.33 

11.22 

8 

.38 

14 

15 

1 12. 14 

8.82 

12.10 

8.87 

12.06 

8.92 

12.02 

8 

.97 

15 

16 

12.94 

9.40 

12.90 

9.46 

12.86 

9.52 

12.82 

9 

.57 

16 

17 

13.75 

9.99 

13.71 

10.05 

13.67 

10.11 

13.62 

10 

. 17 

17 

18 

14.56 

10.58 

14.52 

10.64 

14.47 

10.71 

14.42 

10 

.77 

18 

19 

15.37 

11.17 

15.32 

11.23 

15.27 

11.30 

15.22 

11 

.37 

19 

20 

16.18 

11.76 

16.13 

11.83 

16.08 

11.90 

16.03 

11 

.97 

20 

21 

16.99 

12.34 

16.94 

12.42 

16.88 

12.49 

16.83 

12 

.56 

21 

22 

17. SO 

12.93 

17.74 

13.01 

17.68 

13.09 

17.63 

13 

. 16 

22 

23 

18.61 

13.52 

18.55 

13.60 

18.49 

13.68 

18.43 

13. 

.76 

23 

24 

19.42 

14.11 

19.35 

14.19 

19.29 

14.28 

19.23 

14. 

.36 

24 

25 

120.23 

14.69 

20.16 

14.78 

20.10 

14.87 

20.03 

14. 

,96 

25 

26 

21.03 

15.28 

20 . 97 

15.37 

20.90 

15.47 

20.83 

15. 

,56 

26 

27 

, 21.84 

15.87 

21.77 

15.97 

21.70 

16.06 

21.63  ! 

16. 

, 15 

27 

28 

j 22.65 

16.46 

22.58 

16.56 

22.51 

16.65 

22.44  ! 

16. 

,75 

28 

29 

i 23.46 

i 17.05 

23.39 

17.15 

23.31 

17.25 

23.24  I 

17. 

35 

29 

30  i 

| 24.27 

I 17.63 

24.19 

17.74 

24.12 

17.84 

24.04 

17. 

,95 

30 

'31 

25.08 

! 18.22 

25.00 

18.33 

24.92 

18.44 

24.84 

18. 

,55 

31 

32 

25.89 

! 18.81 

25.81 

18.92 

25.72 

19.03 

25.64  ! 

19. 

15 

32 

33 

26.70 

19.40 

26.61 

19.51 

26.53 

19.63 

26.44  | 

19. 

74 

33 

34 

27.51 

19.98 

! 27.42 

20.10 

27.33 

20.22 

27.24  ! 

20. 

34 

34 

35 

28.32  ! 

20.57 

28.23 

20.70 

28.13 

20.82 

28 . 04 

20. 

94 

35 

36 

29. 12  | 

21.16 

29.03 

21  .29 

28.94 

21.41 

28.85 

21. 

54 

36 

37 

29.93 

21.75 

29.84 

21.88 

29.74 

22.01 

29.65 

22. 

14 

37 

38 

30.74 

22.34 

30.64 

22.47 

30.55 

22.60 

30.45 

22. 

74 

38 

39 

31.55  | 

22.92 

31.45 

23.06 

31.36 

23.20 

31.25 

23. 

33 

39 

40 

32.36  l 

23.51  [ 

32.26 

23.65 

32.15 

23.79 

32.05 

23. 

93 

40 

41 

33.17 

24.10  ! 

33.06 

24.24 

32.96 

24.39 

32.85 

24. 

53 

4l 

42  , 

33.98 

24.69 

33.87 

24.83 

33.76 

24.98 

33.65 

25. 

13 

42 

43 

34.79 

25.27 

34.68 

25.43 

34.57 

25.58 

34.45 

25. 

73 

43 

44  1 

35.60  ; 

25.86  ; 

35.48 

26.02 

35.37 

26.17 

35.26  j 

26. 

33 

44 

45  I 

36.41  ; 

26.45  | 

36.29 

26.61 

36.17 

26.77 

36.06 

26. 

92 

45 

46  1 

37.21 

27.04 

37.  10 

27.20 

36.98 

27.36 

36.86  ! 

27. 

52 

46 

47 

38.02  1 

27.63 

37.90 

27.79 

37.78 

27.96 

37.66 

28. 

12 

17 

48 

38 . 83 

28.21  i 

38.71 

28.38 

18.59 

28 . 55 

38.46 

28. 

72 

46 

19 

39 . 64 

28.80  ! 

39.52 

28.97 

39.39 

29.15 

39.26 

29. 

32 

49 

50 

40.45 

29.39 

40.32 

29.57 

40.10 

29.74 

40.06 

29. 

92 

50 

<o 

Q 

C 

1 

Q 

L 

Dep. 

Lat.  I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Distance. 

54  Deg. 

53}  Deg. 

53*  Deg. 

53*  Dog. 

TRAVERSE  TABLE. 


145 


o 

»■ 

| 36  Deg. 

36*  Dog. 

36*  Deg. 

36*  Deg. 

1, 

S' 

1 ft- 

ps 

3 

§ 

Lttt. 

j Dep. 

Lat. 

Dop. 

Lat 

| Dep 

! Lat. 

Dep. 

3 

8 

51 

52 

53 

54 
66 
66 

67 

68 

59 

60 

141.26 
42  07 
! 42.88 
43.69 
44.50 
45.30 
46.11 
46.92 
47.73 
48.54 

29.98 
30.56 
31.15 
31.74 
32.33 
32.92 
33.50 
34.09 
34.68 
! 35.27 

41.13 

41.94 

42.74 

43.55 

44.35 

45.16 

45.97 

46.77 

47.58 

48.39 

30.16 

30.75 

31.34 

31.93 

32.52 

33.11 

33.70 

34.30 

34.89 

35.48 

41  Too 
41.80 
42.60 
ii  43.41 

1 44.21 

1 45.02 
45.82 
46.62 
47.43 
48.23 

30734 
30.93 
| 31.53 

1 32.12 
32.72 
i 33.31 i 
33.90 
34.50 
35  09 
| 35.69 

40  86 

I 41  67 
! 42.47 

1 43.27 

44.07 
44.87 
45.67 
46.47 
47.27 

48.08 

30.51 
31.11 
31.71 
32.31 
32.91 

33.51 
34.10 
34.70 
35.30 
35  90 

'51 
62 
i 53 

1 54 

! 66 
! 56 
57 
; 58 

59 

60 

61 

49.35 

35.85 

49.19 

36.07 

49.04 

! 36.28 

1 48.88 

36.50 

61 

62 

50.16 

36.44 

50.00 

36.66 

49.84 

j 36.88  j 

49.68 

37.10 

62 

63 

50.97 

37.03 

50.81 

37.25 

50.64 

37.47  ! 

50.48 

37.69 

63 

64 

51.78 

37.62 

51.61 

37.84 

51.45 

! 38.07  ! 

51.28 

38.29 

64 

65 

52.59 

38,21 

52.42 

38.44 

52.25 

38.66  i 

52.08 

38.89 

65 

66 

53.40 

38.79 

53.23 

39.03 

i 53.05 

39.26  ! 

52.88 

t 39.49 

66 

67 

54.20 

39.38 

54.03 

39.62 

53.86 

39.85 

53.68 

40.09 

67 

68 

55.01 

39.97 

54.84 

40.21 

54.66 

40.45  j 

54.49 

40 . 69 

68 

69 

55.82 

40.56 

55.64 

40.80 

55.47 

41.04 

55.29 

41.28 

69 

70 

56.63 

41.14 

56.45 

41.39 

56.27 

41.64 

56.09 

41.88 

70 

71 

57.44 

41.73 

57.26 

41.98 

57.07 

42.23 

56.89 

42.48 

71 

72 

58.25 

42.32 

58.06 

42.57 

57.88 

! 42.83 

57.69 

43.08 

72 

73 

59.06 

42.91 

58.87 

43.17 

58.68 

43.42 

58.49 

43.68 

73 

74 

59.87 

43.50 

59.68 

43.76 

59.49 

44.02 

59.29 

44.28 

74 

75 

60.68 

44.08 

60.48 

44.35 

60.29 

44.61 

60.09 

44.87 

75 

76 

61.49 

44.67 

61.29 

44.94 

61.09 

45.21 

60.90 

45.47 

76 

77 

62.29  I 

45.26 

62.10 

45.53 

61.90 

45.80 

61.70 

46.07 

77 

78 

63.10  ! 

45.85 

62.90 

46.12 

62.70 

46.40 

62.50 

46.67 

78 

79 

63.91 

46.43 

63.71 

46.71 

63.50 

46.99 

63.30 

47.27 

79 

80 

64.72 

47.02 

64.52 

47.30 

64.31 

47.59 

j 64.10 

47.87 

80 

81 

65.53 

47.61 

65.32 

47.90 

65.11  1 

48.18 

64.90 

48.46 

81 

82 

66.34 

48.20 

66.13 

48.49 

65.92 

48.78 

65.70 

49.06 

82 

83 

67.15 

48.79 

66.93 

49.08 

66.72  1 

49.37 

66..  50 

49.66 

83 

84 

67.96 

49.37 

67.74 

49.67 

67.52  | 

49.97 

67.31 

50 . 26 

84 

85 

68.77 

49.96 

68.55 

50.26 

68.33  i 

50.56 

68.11 

50.86 

85 

86 

69.58 

50.55 

69.35 

50.85 

69.13 

51.15 

68.91 

51.46 

86 

87 

70.38 

51. 14  i 

70.16 

51.44 

69  94 

51.75 

69.71 

52.05 

87 

88 

71.19 

51.73 

70.97 

52.04 

70.74 

52.34 

70.51 

52.65 

88 

89 

72.00 

52.31 1 

71  .77 

52.63 

71.54 

52.94 

71.31 

53.25 

89 

90 

72.81 

52.90  i 

72.58 

53.22 

72.35 

53.53 

72.11 

53.85 

90 

91 

73.62 

53.49  1 

73.39 

53.81 

73.15 

54.13 

; 72.91 

54.45 

91 

92 

74.43  | 

54.08  | 

74.19 

54.40 

73.95 

54.72 

73.72 

55.05 

92 

93  t 

75.24 

54.66  ! 

75.00 

54.99 

74.76 

55.32 

74.52 

55.64 

93 

i 94 

76.05 

55.25  ! 

75.81 

55.58 

75.56 

55.91 

75.32 

56 . 24 

94 

95 

76  86  1 

55.84 

76.61 

56.17 

76.37 

56.51 

76.12 

56.84 

95 

96 

77.67  1 

56.43  | 

77.42 

56.7?  I 

77.17 

57.10 

76.92 

57.44 

96 

97 

78.47  ! 

67.02 

78.23 

57.36  I 

77.97 

57.70 

77.72 

58.04 

97 

98  1 

79.26 

57.60 

79.03 

57.95  ! 

78.78 

58.29 

78.52 

58.64 

98 

99 

80.09 

68.19 

79.84 

58.54 

79.58 

58.89 

79- 32 

59.23 

99 

100 

80.90 

68.78 

80.64 

59.13 

80.39 

59.48 

80.J3 

59.83 

100 

c> 

o 

g 

Dep.  | 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

| 

3 

CD  1 

Q 

54  Deg. 

53*  Dog. 

53*  Deg 

53*  Deg. 

3 

CD 

5 

140 


TRAVERSE  TABLE 


0 

1 

37  Deg. 

37 £ Deg. 

37*  Deg. 

37  J Deg. 

C 

S' 

a 

s 

Lat. 

Dep. 

Lat.  1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

§ 

J 

1 

0.80 

0.60 

0.80 

0.61 

0.79 

0.61 

0.79 

IT.eT 

2 

1.60 

1.20 

1.59 

1.21 

1.59 

1.22 

1.58 

1.22 

2 

3 

2.40 

1.81 

2.39 

1.82 

2.38 

1 .83 

2.37 

1.84 

3 

4 

3.19 

2.41 

3. 18 

2.42 

3.17 

2 43 

3.16 

2.45 

4 

5 

3.99 

3.01 

3.98 

3.03 

3.97 

3.04 

3.95 

3.06 

5 

6 

4.79 

3.61 

4.78 

3 63 

4.76 

3.65 

4.74 

3.67 

6 

7 

5.59 

4.21 

5.57 

4.24 

5.55 

4.26 

5.53 

4.29 

7 

8 

6.39 

4.81 

6.37 

4.84 

6.35 

4.87 

6.33 

4.90 

8 

9 

7.19 

5.42 

7. 16 

5 45 

7.14 

5.48 

7.12 

5.51 

9 

10 

7.99 

6.02 

7.96 

6.05 

7.93 

6.09 

7.91 

6.12 

10 

11 

8.78 

6.62 

8.76 

6.66 

8.73 

6.70 

8.70 

6.73 

11 

12 

9.58 

7.22 

9.55 

7.26 

9.52 

7.31 

9.49 

7.35 

12 

13 

10.38 

7.82 

10.35 

7.87 

10.31 

7.91 

10.28 

7.96 

13 

14 

11.18 

8.43 

11.14 

8.47 

11.11 

8.52 

11.07 

8.57 

14 

15 

11.98 

9.03 

11.94 

9.08 

11.90 

9.13 

11.86 

9.18 

15 

16 

12.78 

9.63 

12.74 

9.68 

12.69 

9.74 

12.65 

9.80 

16 

17 

13.58 

10.23 

13.53 

10.29 

13.49 

10.35 

13.44 

10.41 

17 

18 

14.38 

10.83 

14.33 

10.90 

14.28 

10.96 

14.23 

11.02 

18 

19 

15.17 

11.43 

15.12 

11.50 

15.07 

11.57 

15.02 

11.63 

19 

20 

15.97 

12.04 

15.92 

12.11 

15.87 

12.18 

15.81 

12.24 

20 

21 

16.77 

12.64 

16.72 

12.71 

16.66' 

12.78 

16.60 

12.86 

*21 

22 

17.57 

13.24 

17.51 

13.32 

17.45 

13.39 

17.40 

13.47 

22 

23 

18.37 

13.84 

18.31 

13.92 

18.25 

14.00 

18.19 

14.08 

23 

24 

19.17 

14.44 

19. 10 

14.53 

19.04 

14.61 

18.98 

14.69 

24 

25 

19.97 

15.05 

19.90 

15.13 

19.83 

15.22 

19.77 

15.31 

25 

26 

20.76 

15.65 

20.70 

15.74 

20.63 

15.83 

20.56 

15.92 

26 

27 

21.56 

16.25 

21.49 

16.34 

21.42 

16.44 

21.35 

16.53 

27 

28 

22.36 

16.85 

22.29 

16.95 

22.21 

17.05 

22.14 

17.14 

28 

29 

23.16 

17.45 

23.08 

17.55 

23.01 

17.65 

22.93 

17.75 

29 

30 

23.96 

18.05 

23.88 

18.16 

23.80 

18.26 

23.72 

18.37 

30 

31 

24.76 

18.66 

24.68 

18.76 

24.59 

18.87 

24.51 

18.98 

31 

32 

25.56 

19.26 

25.47 

19.37 

25.39 

19.48 

25.30 

19.59 

32 

33 

26.35 

19.86 

26.27 

19.97 

26.18 

20.09 

26.09 

20.20 

33 

34 

27.15 

20.46 

27.06 

20.58 

26.97 

20.70 

26.88 

20.82 

34 

35 

27.95 

21.06 

27.86 

21.19 

27.77 

21.31 

27.67 

21.43 

35 

36 

j 28.75 

21.67 

28.66 

21.79 

28.56 

21.92 

28.46 

22.04 

36 

37 

29.55 

22.27 

29.45 

22.40 

29.35 

22.52 

29.26 

22.65 

37 

38 

30.35 

22.87 

30.25 

23.00 

30.15 

23.13 

30.05 

23.26 

39 

39 

31.15 

23.47 

31.04 

23.61 

30.94 

23.74 

30.84 

23.88 

39 

40 

31.95 

24.07 

31.84 

24.21 

31.73 

24.35 

31.63 

24.49 

40 

41 

32.74 

24.67 

32.64 

24.82 

32  53 

24.96 

32.42 

25.10 

! 41 

42 

33.54 

25.28 

33.43 

25.42 

33  32 

25.57 

33.21 

25.71 

42 

43 

34.34 

25.88 

34.23 

26.03 

34.11 

26.18 

34.00 

26.33 

i *3 

44 

35. 14 

26.48 

35.02 

26.63 

34.91 

26.79 

34.79 

26.94 

44 

45 

35.94 

27  08 

35.82 

27.24 

35.70 

27.39 

35.58 

27.55 

45 

40 

36.74 

27.68 

36.62 

27.84 

36.49 

28.00 

36.37 

28.16 

46 

47 

37.54 

28.29 

37.41 

28.45 

37.29 

28.61 

37.16 

28  77 

1 47 

48 

38.33 

28.89 

38.21 

29.05 

38.08 

29.22 

37.95 

29.39 

i 48 

49 

39. 13 

29.49 

39.00 

29.66 

38.87 

29.83 

38.74 

30.00 

49 

60 

39.93 

30.09 

39.80 

30.26 

39.67 

30.44 

39.53 

30.61 

50 

8 

S 

1 

Dep. 

| Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

s 

53  Dog. 

52}  Dog. 

52*  Dog 

52}  Dog. 

a 

TRAVERSE  TABLE. 


147 


o 

5* 

r Deg. 

37 £ Deg. 

37  i Deg. 

37}  Deg. 

If 

3 

a 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 Dep. 

I 

51 

40.73 

30.69 

40.60 

30.87 

40.46 

31.05 

40.33 

! 31.22 

1 61 

52 

41.53 

31.29 

41.39 

31.48 

41.25 

31.66 

41.12 

31.84 

62 

53 

42.33 

31.90 

42.19 

32.08 

42.05 

32.26 

41.91 

32.45 

53 

54 

43.13 

32.50 

42.98 

32.69 

42.84 

32.87 

42.70 

33.06 

54 

55 

43.92 

33.10 

43.78 

33.29 

43.63 

33.48 

43.49 

33.67 

55 

56 

44.72 

33.70 

44.58 

33.90 

44.43 

34.09 

44.28 

34.28 

56 

67 

45.52 

34.30 

45.37 

34.50 

45.22 

34,70 

45.07 

34.90 

57 

58 

46.32 

34.91 

46.17 

35.11 

46.01 

35.31 

46.86 

35.51 

58 

59 

47.12 

35.51 

46.96 

35.71 

46.81 

35.92 

46.65 

36.12 

59 

60 

47.92 

36.11 

47.76 

36.32 

47.60 

36.53 

47.44 

36.73 

60 

61 

48.72 

36.71 

48.66 

36.92 

48.39 

37.13 

48.23 

37.35 

61 

62 

49.52 

37.31 

49.35 

37.53 

49.19 

37.74 

49.02 

37.96 

62 

63 

50.31 

37.91 

50.16 

38.13 

49.98 

38.35 

49.81 

38.57 

63 

64 

51.11 

38.52 

50.94 

38.74 

50.77 

38.96 

50.60 

39.18 

64 

65 

51.91 

39.12 

51.74 

39.34 

51.57 

39.57 

51.39 

39.79 

65 

66 

52.71 

39.72 

52.54 

39.95 

52.36 

40.18 

52.19 

40.41 

66 

67 

53.61 

40.32 

53.33 

40.55 

53.15 

40.79 

52.98 

41.02 

67 

68 

54.31 

40.92 

54.13 

41.16 

53.95 

41.40 

63.77 

41.63 

68 

69 

55.11 

41.53 

54.92 

41.77 

54.74 

42.00 

54.56 

42.24 

69 

70 

65.90 

42.13 

55.72 

42.37 

55.53 

42.61 

55.35 

42  86 

70 

71 

56.70 

42.73 

56.52 

42.98 

56.33 

43.22 

56.14 

43.47 

'71 

72 

57.50 

43.33 

57.31 

43.58 

57.12 

43.83 

56.93 

44.08 

72 

73 

58.30 

43.93 

58.11 

44.19 

57.91 

44.44 

57.72 

44.69 

73 

74 

59.10 

44.53 

58.90 

44.79 

58.71 

45.05 

58.51 

45.30 

74 

75 

59.90 

45.14 

59.70 

45.40 

59.50 

45.66 

59.30 

45.92 

75 

76 

60.70 

45.74 

60.50 

46.00 

60.29 

46.27 

60.09 

46.53 

76 

77 

61.49 

46.34 

61.29 

46.61 

61.09 

46.87 

60.88 

47.14 

77 

78 

62.29 

46.94 

62.09 

47.21 

61.88 

47.48 

61.67 

47.75 

78 

79 

63.09 

47.54 

62.88 

47.82 

62.67 

48.09 

62.46 

48.37 

79 

80 

63.89 

48.15 

63.68 

48.42 

63.47 

48.70 

63.26 

48.98 

80 

81 

64.69 

48.75 

64.48 

49.03 

64.26 

49.31 

64.05 

49.59 

sr 

82 

65.49 

49.35 

65.27 

49,63 

65.05 

49.92 

64.84 

50.20 

82 

83 

66.29 

49.95 

66.07 

50.24 

65.85 

50.53 

65.63 

50.81 

83 

84 

67.09 

50.55 

66.86 

50.84 

66.64 

51.14 

66.42 

51.43 

84 

85 

67.88 

51.15 

67.66 

51.45 

67.43 

51.74 

67.21 

52.04 

85 

86 

68.68 

51.76 

68.46 

52.06 

68.23 

52.35 

68.00 

52.65 

86 

87 

69.48 

52.36 

69.25 

52.66 

69.02 

52.96 

68.79 

53.26 

87 

88 

70.28 

62.96 

70.05 

53.27 

69.82 

53.57 

69.58 

53.88 

88 

89 

71.08 

63.56 

70.84 

53.87 

70.61 

54.18 

70.37 

54.49 

89 

90 

71.88 

54.16 

71.64 

54.48 

71.40 

54.79 

71.16 

55.10 

90 

91  ! 

72.68 

64.77 

72.44 

55.08 

72.20 

55.40 

71.95 

55.71 

91 

92 

73.47 

55.37 

73.23 

55.69 

72.99 

56.01 

72.74 

56.32 

92 

93 

74.27 

65.97 

74.03 

56.29 

73.78 

56.61 

73 . 53 

56.94 

93 

94 

75.07 

56.57 

74.82 

56.90 

74.58 

57.22 

74.32 

57.55 

94 

95 

75.87 

57.17 

75.62 

57.50 

75.37 

57.83 

75. 12 

58.16 

95 

96 

76.67 

- 57.77 

76.42 

58.11 

76.16 

58.44 

75  91 

68.77 

96 

97 

77 . 47 

58.38 

77.21 

58.71 

76.96 

69.05 

76  70 

59.39 

97 

98 

78.27 

58.98 

78.01 

59.32 

77.75 

59.66 

77.49 

60.00 

98 

99 

79.06 

59.58 

78.80 

59.92 

78.54 

60.27 

78 . 28 

60.61 

99 

lOO 

79.86 

60.18 

79.60 

60.53 

79.34 

60.88 

79.07 

i 61.22 

100 

| 

Dep. 

| Lat. 

Dep. 

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©‘ 

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53  Deg. 

52}  Deg. 

52£  Deg. 

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24 


148 


TRAVKRSE  TABLK 


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5 

m | 

38  Dog. 

38*  Deg. 

38*  Deg. 

38]  Deg. 

71 

3 

2 1 

L*t. 

Dep. 

Lat.  | 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

g 

8 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0.79 

1 58 
2.36 
3.15 
3.94 
4.73 
5.52 
6.30 
7.09 
7.88 

0.62 

1.23 

1 .85 
2.46 
3.08 
3.69 
4.31 
4.93 

5 54 
6.16 

0.79 

1.57 

2.36 

3.14 

3.93 

4.71 

5.50 

6.28 

7.07 

7.85 

0.62 

1.24 

1.86 

2.48 

3.10 

3.71 

4.33 

4.95 

5.57 

6.19 

“0.78 
1.57 
2.35  j 
3.13 
3.91  ! 
4.70 
5.48  ! 
6.26 
7.04  ! 
7.8S  j 

“0 . 62 

1 .24 

1 .87 
2.49 

3 11 
3.74 
4.36 
4.98 
5.60 
6.23 

0.78 

1.56 

2.34 

3.12 

3.90 

4.68 

5.46 

6.24 

7.02 

7.80 

0.63 

1.25 

1.88 

2.50 

3.13 

3.76 

4.38 

5.01 

5 . 63 
0.^0 

1 

2 

3 

4 

5 

3 

7 

8 

0 

13 

11 

8.67 

6.77 

8.64 

6.81 

8.61 

6.85 

8.58 

“6.89 

' 11 

12 

9.46 

7.39 

9.42 

7.43 

9.39 

7.47 

9.36 

7.51 

12 

13 

10.24 

8.00 

10.21 

8.05 

10.17 

8.09 

10.14 

8. 14 

13 

14 

11.03 

8.62 

10.99 

8.67 

10.96 

8.72 

10.92 

8.70 

14 

'5 

11.82 

9.23 

11.78 

9.29 

11.74 

9.34 

11.70 

9.39 

15 

16 

12.61 

9.85 

12.57 

9.91 

12.52 

9.96 

12.48 

10.01 

t 16 

17 

13.40 

10.47 

13.35 

10.52 

13.30 

10.58 

13.26 

10.64 

17 

18 

14.18 

11.08 

14.14 

11.14 

14.09 

11.21 

14.04 

11.27 

18 

19 

14.97 

11.70 

14.92 

11.76 

14.87 

11.83 

14.82 

11.89 

19 

20 

15.76 

12.31 

15.71 

12.38 

15.65 

12.45 

15.60 

12.52 

20 

21 

16.55 

12.93 

16.49 

13.00 

16.43 

13.07 

16  38 

137  14 

21 

22 

17.34 

13.54 

17.28 

13.62 

17.22 

13.70 

17.16 

13.77 

22 

23  | 

18.12 

14.16 

18.06 

14.24 

18.00 

14.32 

17.94 

14.40 

23 

24  j 

18.91 

14.78 

18.85 

14.86  | 

18.78 

14.94  1 

18.72 

15.02 

24 

25 

19.70  1 

15.39 

19.63 

15.48 

19.57 

15.56 

19.50 

15.65 

25 

26 

20.49 

16.01 

20.42 

16.10 

20.35 

16.19  ! 

20.28 

16.27 

26 

27 

21.28 

16.62 

21.20 

16.72 

21.13 

16.81  ! 

121.06 

16.90 

27 

28 

22.06 

17.24 

21.99 

17.33 

21.91 

17.43  j 

;21.84 

17.53 

- 28 

29 

22.85 

17.85 

22.77 

17.95 

22.70 

18.05 

22.62 

18.15 

29 

30 

23.64 

18.47 

| 23.56 

j 18.57 

23.48 

18.68 

j 23.40 

18.78 

30 

31 

24.43 

19.09 

24.34 

19.19 

24.26 

19.30 

24.18 

19.40 

31 

32 

25.22 

19.70 

25.13 

! 19.81 

25.04 

19.92 

24.96 

20 . 03 

32 

33 

; 26.00 

20.32 

25.92 

! 20.43 

25.83 

20.54 

25.74 

20.66 

33 

34 

: 26.79 

20.93 

26.70 

21.05 

26.61 

21.17 

26.52 

21.28 

34 

35 

27.58 

! 21.55 

27.49 

21.67 

27.39 

21.79 

27.30 

21.91 

35 

36 

28.37 

! 22.16 

28.27 

22.29 

28.17 

22.41 

28.08 

22.53 

36 

37 

29.16 

i 22 . 78 

29.06 

22.91 

28.96 

23.03 

28.86 

23.  16 

37 

38 

29.94 

23.40 

29.84 

23.53 

29.74 

23.66 

29.64 

23.79 

38 

39 

30.73 

i 24.01 

30.63 

24.14 

30.52 

24.28 

30.42 

24.41 

39 

40 

31.52 

24.63 

31.41 

24.76 

31.30 

24.90 

31.20 

25.04 

40 

41 

32.31 

25.24 

32.20 

25.38 

32.09 

25.52 

31.98 

25.66 

n 

42 

33.10 

: 25.86 

32.98 

26.00 

32.87 

26.15 

32.76 

26.29 

42 

43 

33.88 

, 26.47 

33.77 

26.62 

33.65 

26.77 

33.53 

26.91 

43 

44 

34.67 

1 27.09 

34.55 

27.24 

34.43 

] 27.39 

34.31 

27.54 

44 

45 

35  46 

j 27.70 

35.34 

27.86 

35.22 

28.01 

35.09  | 28.17 

45 

4£ 

36.25 

28.32 

36.12 

28.48 

36.00 

1 28.64  | 

135.87 

j 28 . 79 

if 

t: 

37.04 

i 28.94 

36.91 

29.10 

36.78 

29  26 

36.65 

! 29.42 

47 

48 

37  82 

29.55 

37.70 

29.72 

37.57 

29  88 

37  43 

30.04 

48 

49 

j 38.61 

30.17 

38.48 

30.34 

38.35 

30.50 

38.21 

30.67 

49 

60 

! 39.40 

30.78 

39.27 

30.95 

39.13 

31__13 

38.93 

31.30 

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8 

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s 

! Dop. 

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1 1 

52  Dog. 

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51}  Deg. 

51  £ Deg. 

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TRAVERSE  TABLE, 


149 


d 

GO* 

38  Deg. 

38$  Deg. 

38$  Deg. 

38|  Deg. 

d 

U) 

P 

o 

CD 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

o 

CD 

~\ 

40.19 

31.40 

40.05 

31.57 

39.91 

31.75 

39.77 

31.92 

51 

52 

40.98 

32.01 

40.84 

32.19 

40.70 

32.37 

40.55 

32.55 

52 

53 

41.76 

32.63 

41.62 

32.81 

41.48 

32.99 

41.33 

33.17 

53 

54 

42.55 

33.25 

42.41 

33.43 

42.26 

33.62 

42.11 

33.80 

54 

55 

43.34 

33.86 

43.19 

34.05 

43.04 

34.24 

42.89 

34.43 

55 

56 

44.13 

34.48 

43.98 

34.67 

43.83 

34.86 

43.67 

35.05 

56 

57 

44.92 

35.09 

44.76 

35.29 

44.61 

35.48 

44.45 

35.68 

57 

58 

45.70 

35.71 

45.55 

35.91 

45.39 

36.11 

45.23 

36.30 

58 

59 

46.49 

36.32 

46.33 

36.53 

46.17 

36.73 

48.01 

36.93 

59 

60 

47.28 

36.94 

47. 12 

37.15 

46.96 

37.35 

46.79 

37.56 

60 

61 

48.07 

37.56 

47.90 

37.76 

47.74 

37.97 

47.57 

38.18 

61 

62 

48.86 

38.17 

48.69 

38.38 

48.52 

38.60 

48.35 

38.81 

62 

63 

49.64 

38.79 

49.47 

39.00 

49.30 

39.22 

49.13 

39.43 

63 

64 

50.43 

39.40 

50.26 

39.62 

50.09 

39.84 

49.91 

40.06 

64 

65 

51.22 

40.02 

51.05 

40.24 

50.87 

40.46 

50.69 

40.68 

65 

66 

52.01 

40.63 

51.83 

40.86 

51.65 

41.09 

51.47 

41. 3 J 

66 

67 

52.80 

41.25 

52.62 

41.48 

52.43 

41.71 

52.25 

41*94 

67 

68 

53.58 

41.86 

53.40 

42.10 

53.22 

42.33 

53.03 

42.56 

68 

69 

54.37 

42.48 

54. 19 

42.72 

54.00 

42.95 

53.81 

43.19 

69 

70 

55.16 

43.10 

54.97 

43.34 

54.78 

43.58 

54.59 

43.81 

70 

71 

55.95 

43.71 

55.76 

43.96 

55.57 

44.20 

55.37 

44  4 

71 

72 

56.74 

44.33 

56.54 

44.57 

56.35 

44.82 

56.  L5 

45.07 

72 

73 

57.52 

44.94 

57.33 

45.19 

57.13 

45.44 

56.93 

45.69 

73 

74 

58.31 

45.56 

58.11 

45.81 

57.91 

46.07 

57.71 

46.32 

74 

75 

59.10 

46. 17 

58.90 

46.43 

58.70 

46.69 

58.49 

46.94 

75 

76 

59.89 

46.79 

59.68 

47.05 

59.48 

47.31 

59.27 

47.57 

76 

77 

60.68 

47.41 

60.47 

47.67 

60.26 

47.93 

60.05 

48.20 

77 

78 

61.46 

48.02 

61.25 

48.29 

61.04 

48.56 

60.83 

48.82 

78 

79 

62.25 

48.64 

62.04 

48.91 

61.83 

49.18 

61.61 

49.45 

79 

80 

63.04 

49.25 

62.83 

49.53 

62.61 

49.80 

62.39 

50.07 

8f) 

81 

63.83 

49.87 

63.61 

50.15 

63.39 

50.42 

63.17 

50.70 

81 

82 

64.62 

50.48 

64.40 

50.77 

64.17 

51.05 

63.95 

51.33 

82 

83 

65.40 

51.10 

65.18 

51.38 

64.96 

51.67 

64.73 

51.95 

83 

84 

66.19 

51.72 

65.97 

52.00 

65.74 

52.29 

65.51 

52.58 

' 84 

85 

66.98 

52.33 

66.75 

52.62 

66.52 

52.91 

66.29 

53.20 

85 

86 

67.77 

52.95 

67.54 

53.24 

67.30 

53.54 

67.07 

53.83 

86 

87 

68.56 

53.56 

68.32 

53.86 

68.09 

54.16 

67.85 

54.46 

87 

88 

69.34 

54. 18 

69.11 

54.48 

68.87 

54.78 

68.63 

55.08 

88 

89 

70.13 

54.79 

69.89 

55.10 

69.65 

55.40 

69.41 

55.71 

89 

90 

70.92 

55.41 

70.68 

55 . 72 

70.43 

56.03 

70.19 

56.33 

90 

91 

71.71 

56.03 

71.46 

56.34 

71.22 

56.65 

70.97 

5G.96 

91 

92 

72.50 

56 . 64 

72.25 

56.96 

72.00 

57.27 

71.75 

57.58 

92 

93 

73.28 

57.26 

73.03 

57.58 

72.78 

57.89 

72.53 

58.21 

93 

94 

74.07 

57.87 

73.82 

58.19 

73.57 

58.52 

73.31 

58.84 

94 

95 

74.86 

58.49 

74.61 

58.81 

74.35 

59.14 

74.09 

59.46 

95 

96 

75.65 

59.10 

75.39 

59.43 

75.13 

59.76 

74.87 

60.09 

96 

97 

76.44 

59.72 

76.18 

60.05 

75.91 

60.38 

75.65 

60.71 

97 

98 

77.22 

60.33 

76 . 96 

60.67 

76.70 

61.01 

76.43 

61.34 

98 

99 

78.01 

60.95 

77.75 

61.29 

77.48 

61.63 

77.21 

61.97 

99 

100 

78.80 

61.57 

78.53 

61.91 

78.26 

62.25 

77.99 

62.59 

100 

a> 

o 

C 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

c 

ri 

tn 

3 

52  Deg. 

51|  Deg. 

51$  Deg. 

51$  Deg. 

gJ 

5 

150 


TRAVERSE  TA11LR. 


1 

IT 

£ 

| 39  Dog. 

394  Deg. 

39  \ 

Deg. 

39*  Deg. 

5 

u> 

a 

B 

S 

Lat. 

* Dep. 

i 

Lat. 

Dep. 

Lat. 

1 Dop. 

Lat. 

| Dep. 

a 

o 

8 

l 

0.78 

0.63 

0.77 

0.63 

0.77 

0.64 

0.77 

0 64 

1 

2 

1 .56 

1.26 

1.65 

1.27 

1.64 

1.27 

1.54 

1.28 

2 

3 

2.33 

1.89 

2.32 

1.90 

2.31 

I 1.91 

2.31 

1.92 

3 

4 

3.11 

2.62 

3.10 

2.53 

j 3.09 

| 2.64 

1 3.08 

2.66 

4 

5 

3.89 

3.15 

3.87 

3.16 

3.86 

3.18 

3.84 

3.20 

6 

6 

4.66 

3.78 

4.65 

3.80 

|l  4.63 

3.82 

4.61 

3.84 

6 

7 

j 6.44 

4.41 

5.42 

4.43 

j 5.40 

4.45 

5.38 

4.48 

7 

8 

6.22 

5.03 

6.20 

5.06 

j!  6.17 

5.09 

6.  15 

5.12 

8 

9 

6.99 

6.66 

6.97 

6.69 

6.94 

6.72 

6.92 

5.75 

9 

10 

7.77 

6.29 

7.74 

6.33 

1 7.72 

6.36 

7.69 

6.39 

10 

11 

8.55 

6.92 

8.62 

6.96 

8.49 

“ 7.00 

8.46 

7.03 

11 

12 

9.33 

7.55 

9.29 

7.59 

I!  9.26 

7.63 

9.23 

7.67 

12 

13 

10.10 

8.18 

10.07 

8.23 

; 10.03 

8.27 

9.99 

8.31 

13 

14 

10.88 

8.81 

10.84 

8.86 

10.80 

8.91 

10.76 

8.95 

14 

15 

11.66 

9.44 

11.62 

9.49 

j 11.57 

9.54 

11.53 

9.59 

J 5 

16 

12.43 

10.07 

12.39 

10.12 

' 12.35 

10.18 

12.30 

10.23 

16 

17 

13.21 

10.70 

13.16 

10.76 

j 13.12 

10.81 

13.07 

10.87 

17 

18 

13.99 

11.33 

13.94 

11.39 

1 13.89 

11.45 

13.84 

11.51 

18 

19 

14.77 

11.96 

14.71 

12.02 

! 14  66 

12.09 

14.61 

12.15 

19 

20 

15.54 

12.59 

16.49 

12.65 

| 15.43 

12.72 

15.38 

12.79 

20 

21 

16.32 

13.22 

16.26 

13.29 

16.20 

13.36 

16.15 

13.43 

21 

22 

17.10 

13.84 

17.04 

13.92 

16.98 

13.99 

16.91 

14.07 

22 

23 

I 17.87 

14.47 

17.81 

14.55 

17.75 

14.63 

17.68 

14.71 

23 

24 

1 18.65 

15. 10 

18.59 

15.18 

18.52 

15.27 

18.45 

15.35 

24 

25 

i 19.43 

j 15.73 

19.36 

15.82 

19.29 

15.90 

19.22 

15.99 

25 

26 

; 20.21 

16.36 

20.13 

16.45 

20.06 

16.54 

19.99 

16.63 

26 

27 

20.98 

16.99 

20.91 

17.08 

20.83 

17.17 

20.76 

17.26 

27 

38 

21.76 

17.62 

21.68 

17.72 

21.61 

17.81 

21.53 

17.90 

28 

29 

22.54 

18.25 

22.46 

18.35 

22.38 

18.45 

22.30 

18.54 

29 

30 

23.31 

18.88 

23.23 

18.98 

23.15 

19.08 

23.07 

19.18 

30 

'31 

24.09 

19.51 

24.01 

19.61 

23.92 

19.72 

23.83 

19.82 

31 

32  ; 

24.87 

20.14 

24.78 

20.25 

24.69 

20.35 

24.60 

20.46 

32 

33 

25.65 

20.77 

25.55 

20.88 

25.46 

20.99 

25.37 

21.10 

33 

34 

26.42 

21.40 

26.33 

21.51 

26.24 

21.63 

26.14 

21.74 

34 

35 

27.20 

22.03 

27.10 

22.14 

27.01 

22.26 

26.91 

22.38 

35 

36 

27.98 

22.66 

27.88 

22.78 

27.78 

22.90 

27.68 

23.02 

36 

37 

28.75 

23.28 

28.65 

23.41 

28.55 

23.53 

28.45 

23.66 

37 

38 

29.53 

23.91 

29.43 

24.04 

29.32 

24.17 

29.22 

24.30 

33 

39 

30.31 

24 . 54 

30.20 

24.68 

30.09 

24.81 

29.98 

24.94 

39 

40 

31.09 

25.17 

30.98 

25.31 

30.86 

25.44 

30.75 

25.58 

40 

41 

31.86 

25.80 

31.75 

25.94 

31.64 

26.08 

31.52 

26.22 

41 

42  j 

32.64 

26.43 

32 . 52 

26 . 57 

32.41 

26.72 

32.29 

26.86 

42 

43  | 

33.42 

27.06 

33.30 

27.21 

33.18 

27.35 

33.06 

27.50 

43 

44 

34.19 

27.69 

34.07 

27.84 

33.95 

27.99 

33.83 

28.14 

44 

45 

34.97 

28.32 

34.85 

28.47 

34.72 

28.62 

34.60 

28.77 

45 

16 

35.76 

28.95 

35.62 

29.10 

35.49 

29.26 

35.37 

29.41 

46 

47  ! 

36.53 

29.68 

36.40 

29.74 

36.27 

29.90 

36.14 

30.05 

47 

18 

37.30 

30.21 

37.17 

30.37 

37.04 

30.53 

36.90 

30.69 

48 

49 

38.08 

30.84 

37.95 

31.00 

37.81  131.17 

37.67 

31.33 

49 

60 

38.80 

31 .47  1 

38.72 

31.64 

38.58 

31.80 

38.44 

31.97 

60 

8 

Dep.  | 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

9 

Q 

51  Deg. 

50J  Dog. 

50J  Dog. 

60*  Dog. 

5 

00 

5 

o 

S' 

I 

51 

52 

63 

54 

56 

56 

57 

68 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88  j 

89 

90  ! 

91  | 

92 

93 

94 

95 

96  i 

97 

98 

99  | 

J00 

o'  I 

o I 

§ 1 

« I 

S ! 


traverse  table. 


151 


Jos. 


Dep. 


32.10 
32.72 
33.35 
33.98 
34  61 
35.24 
35.87 
36.50 
37.13 
37.76 
'38.39 
39.02 
39.65 
40.28 
40.91 
41.54 
42.16 
42.79 
43.42 
44.05 


44.68 

45.31 

45.94 

46.57 

47.20 

47.83 

48.46 

49.09 

49.72 

50.35 


50.97 
51  .60 
52.23 
52.86 
53.49 
54.12 
54.75 
55.38 
56.01 
56.64 
57.27 
57.90 
58.53 
59.16 
59.79 
60.41 
61.04 
61.67 
62.30 
62.93 

Lat. 


►eg. 


39}  Deg. 

39}  Deg. 

39}  Dog. 

| Distance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

39.49 

32.27 

39.35 

32.44 

39.21 

32.61 

51 

40.27 

32.90 

40.12 

33.08 

39.98 

33  25 

52 

41.04 

33.53 

40.90 

33.71 

40.75 

33  89 

53 

41.82 

34.17 

41.67  l 34.35 

41.52 

34.53 

54 

42.59 

34.80 

42.44 

34.98 

42.29 

35.1? 

55 

43.37 

35.43 

43.21 

35.62 

43.06 

35.81 

56 

44.14 

36.06 

43.98 

36.26 

43.82 

36.45 

57 

44.91 

36.70 

44.75 

36.89 

44.59 

37.09 

58 

45.69 

37.33 

45.53 

37.63 

45.36 

37.73 

59 

46.46 

37.96 

46.30 

38.16 

46.13 

38.37 

60 

47.24 

38.60 

47.07 

38  80 

46.90 

39.01 

61 

48.01 

39.23 

47.84 

34.44 

47.67 

39.65 

62 

48.79 

39.86 

48.61 

40.07 

48.44 

40.28 

63 

1 49.56 

40.49 

49.38 

40.71 

49.21 

40.92 

64 

50.34 

41.13 

50.16 

41.35 

49.97 

41.56 

65 

51.11 

41.76 

50.93 

41.98 

50.74 

42.20 

66 

51.88 

42.39 

51.70 

42.62 

51.51 

42.84 

67 

52.66 

43.02 

52.47 

43.25 

52.28 

43.48 

68 

53.43 

43.66 

53.24 

43.89 

53.05 

44.12 

69 

54.21 

44.29 

54.01 

4.53 

53.82 

44.76 

70 

54.98 

44.92 

54.79 

45.16 

54.59 

45.40 

71 

55.76 

45.55 

55.56 

45.80 

55.36 

46.04 

72 

56.53 

46.19 

56.33 

46.43 

56.13 

46.68 

73 

57.31 

46.82 

57.10 

47.07 

56.89 

47.32 

74 

58.08 

47.45 

57.87 

47.71 

57.66 

47.96 

75 

58.85 

48.09 

58.64 

48.34 

58.43 

48.60 

76 

59.63 

48.72 

59.42 

48.98 

59.20 

49.24 

77 

60.40 

49.35 

60.19 

49.61 

59.97 

49.88 

78 

61.18 

49.98 

60.96 

50.25 

60.74 

50.52 

79 

61.95 

50.62 

61.73 

50.89 

61.51 

51.16 

80 

62.73 

51.25 

62.50 

51.52 

62.28 

51.79 

*81 

63.50 

51.88 

63.27 

52.16 

63.04 

52.43 

82 

64.27 

52.51 

64.04 

52.79 

63.81 

53.07 

83 

65.05 

53.15 

64.82 

53.43 

64.58 

53.71 

84 

65.82 

53.78 

65.59 

54.07 

65.35 

54.35 

85 

66.60 

54.41 

66.36 

64.70 

66.12 

54.99 

86 

67.37 

55.05 

67.13 

55.34 

66.89 

55.63 

87 

68.15 

55.68 

67.90 

55.97 

67.66 

56.27 

88 

68.92 

56.32 

68.67 

56.61 

68.43 

56.91 

89 

1 69.70 

56.94 

69.45 

57.25 

69.20 

57.55 

90 

| 70.47 

57.58 

70.22 

57.88 

69.96 

58.19 

1 91 

71.24 

58.21 

70.99 

58.52 

70.73 

58.83 

1 92 

72.02 

58.84 

71 .76 

59.16 

7i.50 

59.4“ 

93 

72.79 

59.47 

72.53 

59.79 

72.27 

60.11 

94 

73.57 

60.1 1 

73.30 

60.43 

73.04 

60.75 

96 

74.34 

60.74 

74.08 

61.06 

73.81 

61.39 

96 

I 75. IS 

61.37 

74.85 

61.70 

74.58 

62.03 

97 

| 75.89 

62.01 

75.62 

62.34 

75.35 

62.66 

98 

; 76.66 

62.64 

76.39 

62.97 

76.12 

63.30 

99 

77.44 

63.27^ 

77.16 

63.61 

76.88 

63.94 

100 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 1 

50}  Deg. 

50}  Deg 

50}  Deg. 

a 

i 

152 


TRAVERSE  TABLE. 


O 

tn 

40  Deg. 

40J  Deg. 

40 £ Deg. 

40 1 Deg. 

C 

cn* 

P 

3 

O 

n> 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D 

O 

<0 

1 

0.77 

0.64 

0.76 

0.65 

0.76 

0.65 

0.76 

0.65 

1 

2 

1.53 

1.29 

1.53 

1.29 

1.52 

1.30 

1.52 

1.31 

2 

3 

2.30 

1.93 

2.29 

1.94 

2.28 

1.95 

2.27 

1 .96 

3 

4 

3.06 

2.57 

3.05 

2.58 

3.04 

2.60 

3.03 

2.61 

4 

5 

3.83 

3.21 

3.82 

3.23 

3.80 

3.25 

3.79 

3.26 

6 

6 

4.60 

3.80 

4.58 

3.88 

4.56 

3.90 

4.55 

3-92 

6 

7 

5.36 

4.50 

5.34 

4.52 

5.32 

4.55 

5.30 

4.57 

7 

8 

6.13 

5.14 

6.11 

5.17 

6.08 

5.20 

6.06 

5.22 

8 

9 

6.89 

5.79 

6.87 

5.82 

0.84 

5.84 

6.82 

5.87 

9 

10 

7.66 

6.43 

-7.63 

6.46 

7.60 

6.49 

7.58 

6.53 

10 

11 

8.43 

7.07 

8.40 

7.11 

8.36 

7.14 

8.33 

7.18 

11 

12 

9.19 

7.71 

9.16 

7.75 

9.12 

7.79 

9.09 

7.83 

12 

13 

9.96 

8.36 

9.92 

8.40 

9.89 

8.44 

9.85 

8.49 

13 

14 

10.72 

9.00 

10.69 

9.05 

10.65 

9.09 

10.61 

9.14 

14 

15 

11.49 

9. 64 

11.45 

9.69 

11.41 

9.74 

11.36 

9.79 

15 

16 

12.26 

10.28 

12.21 

10.34 

12.17 

10.39 

12.12 

10.44 

16 

17 

13.02 

10.93 

12.97 

10.98 

12.93 

11.04 

12.88 

11 .10 

17 

18 

13.79 

11.57 

13.74 

11.63 

13.69 

11.69 

13.64 

11.75 

18 

19 

14.55 

12.21 

14.50 

12.28 

14.45 

12.34 

14.39 

12.40 

19 

20 

15.32 

12.86 

15.26 

12.92 

15.21 

12.99 

15.15 

13.06 

20 

21 

16.09 

13.50 

16.03 

13.57 

15.97 

13.64 

15.91 

13.71 

21 

22 

16.85 

14.14 

16.79 

14.21 

16.73 

14.29 

16.67 

14.36 

22 

23 

17.62 

14.78 

17.55 

14.86 

17.49 

14.94 

17.42 

15.01 

23 

24 

18.39 

15.43 

18.32 

15.51 

18.25 

15.59 

18.18 

15.67 

24 

25 

19.15 

16.07 

19.08 

16.15 

19.01 

16.24 

18.94 

16.32 

25 

26 

19.92 

16.71 

19.84 

16.80 

19.77 

16.89 

19.70 

16.97 

26 

27 

20.68 

17.36 

20.61 

17.45 

20.53 

17.54 

20.45 

L7.62 

27 

28 

21.45 

18.00 

21.37 

18.09 

21.29 

18.18 

21.21 

18.28 

28 

29 

22.22 

18.64 

22.13 

18.74 

22.05 

18.83 

21.97 

18.93 

29 

30 

22.98 

19.28 

22.90 

19.38 

22.81 

19.48 

22.73 

19.58 

30 

31 

23.75 

19.93 

23.66 

20.03 

23.57 

20.13 

23.48 

20.24 

31 

32 

24.51 

20.57 

24.42 

20.68 

24.33 

20.78 

24.24 

20.89 

32 

33 

25.28 

21.21 

25.19 

21.32 

25.09 

21.43 

25.00 

21.54 

33 

34 

26.05 

21.85 

25.95 

21.97 

25.85 

22.08 

25.76 

22.19 

34 

35 

26.81 

22.50 

26.71 

22.61 

26.61 

22.73 

26.51 

22.85 

35 

30 

27.58 

23.14 

27.48 

23.26 

27.37 

23.38 

27.27 

23.50 

36 

37 

28.34 

23.78 

28.24 

23.91 

28.13 

24.03 

28.03 

24.15 

37 

38 

29.11 

24.43 

29.00 

24.55 

28.90 

24.68 

28.79 

24.80 

38 

39 

29.88 

25.07 

29.77 

25.20 

29.66 

25.33 

29.54 

25.46 

39 

40 

30.64 

25.71 

30.53 

25.84 

30.42 

25.98 

30.30 

26.11 

40 

41 

31.41 

26.35 

31.29 

26.49 

31.18 

26.63 

31.06 

26.76 

41 

42 

32.17 

27.00 

32.06 

27.14 

31.94 

27.28 

31.82 

27.42 

42 

43 

32.94 

27.64 

32.82 

27.78 

32.70 

27.93 

32.58 

28.07 

43 

44 

33.71 

28.28 

33.58 

28.43 

33.46 

28.58 

33.33 

28.72 

44 

45 

34.47 

28.93 

34.35 

29.08 

34.22 

29.23 

34.09 

29.37 

45 

46 

35.24 

29.57 

35.11 

29.72 

34.98 

29.87 

34.85 

30.03 

46 

47 

36.00 

30.21 

35.87 

30.37 

35.74 

30.52 

35.61 

30.68 

47 

48 

36.77 

30.85 

36.64 

31.01 

36.50 

31.17 

36.36 

31.33 

48 

49 

37.54 

31.50 

37.40 

31.66 

37.26 

31.82 

37.12 

31.99 

49 

50 

38.30 

32.14 

38.16 

32.31 

38.02 

32.47 

37.88 

32.64 

50 

a> 

o 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

d 

o 

c 

ed 

(/> 

5 

50  Deg. 

49!  Deg. 

49£  Deg. 

49£  Deg. 

rt 

Til  AVERSE  TABLE. 


153 


o 

w 

40  Deg. 

404  Deg. 

40£  Deg. 

40|  Deg. 

O 

CO 

P 

a 

o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

n 

p 

51 

39.07 

32.78 

38.92 

32.95 

38.78 

33.12 

38.64 

33.29 

Hi 

52 

39.83 

33.42 

39.69 

33.60 

39.54 

33.77 

39.39 

33.94 

52 

53 

40.60 

34.07 

40.45 

34.24 

40.30 

34.42 

40.15 

34.60 

53 

54 

41.37 

34.71 

41.21 

34.89 

41.06 

35.07 

40.91 

35.25 

54 

55 

42.13 

35.35 

41.98 

35.54 

41.82 

35.72 

41.67 

35.90 

55 

56 

42.90 

36.00 

42.74 

36.18 

42.58 

36.37 

42.42 

36.55 

56 

57 

43.66 

36.64 

43.50 

36.83 

43.34 

37.02 

43.18 

37.21 

57 

58 

44.43 

37.28 

44.27 

37.48 

44.10 

37.67 

43.94 

37.86 

58 

59 

45.20 

37.92 

45.03 

38.12 

44.86 

38.32 

44.70 

38.51 

59 

60 

45.96 

38.57 

45.79 

38.77 

45.62 

38.97 

45.45 

39.17 

60 

61 

46.73 

39.21 

46.56 

39.41 

46.38 

39.62 

46.21 

39.82 

61 

62 

47.49 

39.85 

47.32 

40.06 

47.15 

40.27 

46.97 

40.47 

62 

63 

48.26 

40.50 

48.08 

40.71 

47.91 

40.92 

47.73 

41.12 

63 

64 

49.03 

41.14 

48.85 

41.35 

48.67 

41.56 

48.48 

41.78 

64 

65 

49 . 79 

41.78 

49.61 

42.00 

49.43 

42.21 

49.24 

42.43 

65 

66 

50.56 

42.42 

50.37 

42.64 

50.19 

42.86 

50.00 

43.08 

66 

67 

51.32 

43.07 

51.14 

43.29 

50.95 

43.51 

50.76 

43.73 

67 

68 

52.09 

43.71 

51.90 

43.94 

51.71' 

44.16 

51.51 

44.39 

68 

69 

52.86 

44.35 

52.66 

44.58 

52.47 

44.81 

52.27 

45.04 

69 

70 

53.62 

45.00 

53.43 

45.23 

53.23 

45.46 

53.03 

45.69 

70 

71 

54.39 

45.64 

54.19 

45.87 

53.99 

46.11 

53.79 

46.35 

71 

72 

55.16 

46.28 

54.95 

46.52 

54.75 

46.76 

54.54 

47.00 

72 

73 

55.92 

46.92 

55.72 

47.17 

55.51 

47,41 

55.30 

47.65 

73 

74 

56.69 

47.57 

56.48 

47.81 

56.27 

48.06 

56.06 

48.30 

74 

75 

57.45 

48.21 

57.24 

48.46 

57.03 

48.71 

56.82 

48.96 

75 

76 

58.22 

48.85 

58.01 

49.11 

57.79 

49.36 

57.57 

49.61 

76 

77 

58.99 

49.49 

58.77 

49.75 

58.55 

50.01 

58.33 

50.26 

77 

78 

59.75 

50.14 

59.53 

50.40 

59.31 

50.66 

59.09 

50.92 

78 

79 

60  ..52 

50.78 

60.30 

51.04 

60.07 

51.31 

59.85 

51.57 

79 

80 

61.28 

51.42 

61.06 

51.69 

60.83 

51.96 

60.61 

52.22 

80 

81 

62.05 

52.07 

61.82 

52.34 

61.59 

52.61 

61.36 

52.87 

81 

82 

62.82 

52.71 

62.59 

52.98 

62.35 

53.25 

62.12 

53.53 

82 

83 

63.58 

53.35 

63.35 

53.63 

63.11 

53.90 

62.88 

54.18 

83 

84 

64.35 

53.99 

64.11 

54.27 

63.87 

54.55 

63.64 

54.83 

84 

85 

65.11 

54.64 

64.87 

54.92 

64  63 

55.20 

64.39 

55.48 

85 

86 

65.88 

55.28 

65.64 

55.57 

65  39 

55.85 

65.15 

56.14 

86 

87 

66.65 

55.92 

66.40 

56.21 

66  16 

56.50 

65.91 

56.79 

87 

88 

67.41 

56.57 

67.16 

56.86 

66  92 

57.15 

66.67 

57.44 

88 

89 

68.18 

57.21 

67.93 

57.50 

67  68 

57.80 

67.42 

58.10 

89 

90 

68.94 

57.85 

68.69 

58.15 

68.44 

58.45 

68.18 

58.75 

90 

91 

69.71 

58.49 

69.45 

58.80 

69.20 

59.10 

68.94 

59.40 

91 

92 

70.48 

59.14 

70.22 

59.44 

69.96 

59.75 

69.70 

60.05 

92 

93 

71.24 

59.78 

70.98 

60.09 

70.72 

60.40 

70.45 

60.71 

93 

94 

72.01 

60.42 

71.74 

60.74 

71.48 

61.05 

71.21 

61.36 

94 

95 

72,77 

61.06 

72.51 

61.38 

72.24 

61.70 

71.97 

62.01 

95 

96 

73.54 

61.71 

73.27 

62.03 

73.00 

62.35 

72.73 

62.66 

96 

97 

74.31 

62.35 

74.03 

62.67 

73.76 

63.00 

73.48 

63.32 

97 

98 

75.07 

62.99 

74.80 

63.32 

74.52 

63.65 

74.24 

63.97 

98 

99 

75.84 

63.64 

75.56 

63.97 

75:28 

64.30 

75.00 

64.62 

99 

100 

76.60 

64.28 

76.32 

64.61 

76.04 

64.94 

75.76 

65.28 

100 

6 

o 

s 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

o 

C 

is 

.2 

D 

50  Deg. 

49|  Deg. 

49  $ Deg. 

49 i Deg. 

cj 

CO 

Q 

154 


TRAVERSE  TABLE. 


£ 

t/>* 

41  Deg. 

41$  Deg. 

411 

Deg. 

41$  Deg. 

O 

p 

B 

s 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

a 

n 

a 

I 

0.75 

0.66 

0.75 

0.66 

0.75 

0.66 

0.75 

0.67 

i 

o 

1.61 

1.31 

1.60 

1.32 

1.50 

1.33 

1.49 

1.33 

2 

3 

2.26 

1.97 

2.26 

1 .98 

2.25 

1 .99 

2.24 

2.00 

3 

4 

3.02 

2.62 

3.01 

2.64 

3!oo 

2.65 

2.98 

2.66 

4 

5 

3.77 

3.28 

3.76 

3.30 

3.74 

3.31 

3.73 

3.33 

5 

6 

4.53 

3.94 

4.51 

3.96 

4.49 

3.98 

4.48 

4.00 

6 

T 

5.28 

4.59 

5.26 

4.62 

5.24 

4.64 

5.22 

4.66 

7 

8 

6.04 

5.25 

6.01 

5.27 

6 99 

5.30 

5.97 

5.33 

8 

9 

6.79 

5.90 

6.77 

5.93 

6.74 

5.96 

6.71 

5.99 

9 

10 

7.55 

6.56 

-7.52 

6.59 

7.49 

6.63 

7.46 

6.66 

10 

11 

8.30 

7.22 

8.27 

7.25 

8.24 

7.29 

8.21 

7.32' 

11 

12 

9.06 

7.87 

9.02 

7.91 

8.99 

7.95 

8.95 

7.99 

13 

13 

9.81 

8.53 

9.77 

8.57 

9.74 

8.61 

9.70 

8.66 

13 

14 

10.57 

9.18 

10.53 

9.23 

10.49 

9.28 

10.44 

9.32 

14 

15 

11.32 

9.84 

11.28 

9.89 

11.23 

9.94 

11.19 

9.99 

15 

10 

12.08 

10.50 

12.03 

10.55 

11.98 

10.60 

11.94 

10.65 

16 

17 

12.83 

11.15 

12.78 

11.21 

12.73 

11.26 

12.68 

11.32 

17 

18 

13.58 

11.81 

13.53 

11.87 

13.48 

11.93 

13.43 

1 1 . 99 

18 

19 

14.34 

12.47 

14.28 

12.53 

14.23 

12.59 

14.18 

12.65 

19 

20 

15.09 

13. 12 

15.04 

13.19 

14.98 

13.25 

14.92 

13.32 

20 

21 

15.85 

13.78 

15.79 

13.85 

15.73 

13.91 

15.67 

13.98 

21 

22 

16.60 

14.43 

16.54 

14.51 

16.48 

14.58 

16.41 

14.65 

22 

23 

17.36 

15.09 

17.29 

15.16 

17.23 

15.24 

17.16 

15.32 

23 

24 

18.11 

15.75 

18.04 

15.82 

17.97 

15.90 

17.91 

15.98 

24 

25 

18.87 

16.40 

18.80 

16.48 

18.72 

16.57 

18.65 

16.65 

25 

26 

19.62 

17.06 

19.55 

17.14 

19.47 

17.23 

19.40 

17.31 

26 

27 

20.38 

17.71 

20.30 

17.80 

20.22 

17.89 

20.14 

17.98 

27 

28 

21.13 

18.37 

21.05 

18.46 

20.97 

18.55 

20.89 

18.64 

28 

29 

21.89 

19.03 

21.80 

19.12 

21.72 

19.22 

21.64 

19.31 

29 

30 

22.64 

19.68 

22.56 

19.78 

22.47 

19.88 

22.38 

19.98 

30 

31 

23.40 

20.34 

23.31 

20.44 

23.22 

20.54 

23. 13 

20.64 

31 

32 

24.15 

20.99 

24.06 

21.10 

23.97 

21.20 

23.87 

21.31 

32 

33 

24.91 

21.65 

24.81 

21.76 

24.72 

21 .87 

24.62 

21.97 

33 

34 

25.66 

22.31 

25.56 

22.42 

25.46 

22.53 

25.37 

22.64 

34 

35 

26.41 

22.96 

26.31 

23.08 

26.21 

23.19 

26.11 

23.31 

35 

36 

27.17 

23.62 

27.07 

23.74 

26.96 

23.85 

26.86 

23.97 

36 

37 

27.92 

24.27 

27.82 

24.40 

27.71 

24.52 

27.6.0 

24.64 

37 

38 

28.68 

24.93 

28.57 

25.06 

28.46 

25.18 

28.35 

25.30 

.38 

39 

29.43 

25.59 

29.32 

25.71 

29.21 

25.84 

29.10 

25.97 

39 

40 

30.19 

26.24 

30.07 

26.37 

29.96 

26.50 

29.84 

26.64 

40 

41 

30.94 

26.90 

30.83 

27.03 

30.71 

27.17 

30.59 

27.30 

41 

42 

31.70 

27.55 

31.58 

27.69 

31.46 

27.83 

31.33 

27.97 

42 

43 

32.45 

28.21 

32.33 

28.35 

32.21 

28.49 

32.08 

28.63 

43 

44 

33.21 

28.87 

33.08 

29.01 

32.95 

29.16 

32.83 

29.30 

44 

45 

33.96 

29.52 

33.83 

29.67 

33-70 

29.82 

33.57 

29.97 

45 

46 

34.72 

30.18 

34.58 

30.33 

34.45 

30.48 

34.32 

30.63 

46 

47 

35.47 

30.83 

35.34 

30.99 

35.20 

31.14 

35.06 

31.30 

47 

48 

36.23 

31.49 

36.09 

31.65 

35.95 

31.81 

35.81 

31.96 

48 

49 

36.98 

32.15 

36.84 

32.31 

36.70 

32.47 

36.56 

32.63 

49 

60 

37.74 

32.80 

37.59 

32.97 

37.45 

33.13 

37.30 

33.29 

50 

<D 

U 

C 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

o 

c 

«3 

•/) 

Q 

49  Degf 

48f  Deg. 

48$  Deg. 

48$  Deg. 

d 

n 

Q 

TRAVERSE  TABLE. 


155 


o 

sr 

41  Deg. 

4H  Deg. 

41$  Deg. 

41 1 Deg. 

O 

Q) 

§ 

o 

? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

o 

a 

"61 

38.49 

33.46 

38.34 

33.63 

38.20 

33.79 

38.05 

33.96 

51 

52 

39.24 

34.12 

39.10 

34.29 

38.95 

34.46 

38.79 

34.63 

52 

53 

40.00 

34.77 

39.85 

34.95 

39.69 

35. 12 

39.54 

35.29 

53 

54 

40.75 

35.43 

40.60 

35.60 

40.44 

35.78 

40.29 

35.96 

54 

55 

41.51 

36.08 

41.35 

36.26 

41.19 

36.44 

41.03 

36.62 

55 

56 

42.26 

36.74 

42.10 

36.92 

41.94 

37.11 

41.78 

37.29 

56 

57 

43.02 

37.40 

42.85 

37.58 

42.69 

37.77 

42.53 

37.96 

57 

58 

43.77 

38.05 

43.61 

38.24 

43.44 

38.43 

43.27 

38.62 

58 

59 

44.53 

38.71 

44.36 

38.90 

44.19 

39.09 

44.02 

39.29 

59 

(50 

45.28 

39.36 

45.11 

39.56 

44.94 

39.76 

44.76 

39.95 

60 

61 

46.04 

40.02 

45.86 

40.22 

45.69 

40.42 

45.51 

40.62 

61 

62 

46.79 

40.68 

46.61 

40.88 

46.44 

41.08 

46.26 

41.28 

62 

63 

47.55 

41.33 

47.37 

41.54 

47.18 

41.75 

47.00 

41.95 

63 

64 

48.30 

41.99 

48.12 

42.20 

47.93 

42.41 

47.75 

42.62 

64 

65 

49.06 

42.64 

48.87 

42.86 

48.68 

43.07 

48.49 

43.28 

65 

66 

49.81 

43.30 

49.62 

43.52 

49.43 

43.73 

49.24 

43.95 

66 

67 

50.57 

43.96 

50.37 

44. 18 

50.18 

44.40 

49.99 

44.61 

67 

68 

51.32 

44. 6J 

51.13 

44.84 

50.93 

45.06 

50.73 

45.28 

68 

69 

52.07 

45.27 

51.88 

45.49 

51.68 

45.72 

51.48 

45.95 

69 

70 

52.83 

45.92 

52.63 

46 . 1 5 

52.43 

46.38 

52.22 

46.61 

70 

71 

53.58 

46.58 

53.38 

46.81 

53.18 

47.05 

52.97 

47.28 

71 

72 

54.34 

47.24 

54.13 

47.47 

53.92 

47.71 

53.72 

47.94 

72 

73 

55.09 

47.89 

54.88 

48.13 

54.67 

48.37 

54.46 

48.61 

73 

74 

55.85 

48.55 

55.64 

48.79 

55.42 

49.03 

55.21 

49.28 

74 

75 

56.60 

49.20 

56.39 

49.45 

56.17 

49.70 

55.95 

49.94 

75 

76 

57.36 

49.86 

57.14 

50.11 

56.92 

50.36 

56.70 

50.61 

76 

77 

58.1! 

50.52 

57.89 

50.77 

57.67 

51.02 

57.45 

51.27 

77 

78 

58.87 

51.17 

58 . 64 

51.43 

58.42 

51.68 

58.19 

51.94 

78 

79 

59.62 

51.83 

59.40 

52.09 

59.17 

52.35 

58.94 

52.60 

79 

80 

60.38 

52.48 

60.15 

52.75 

59.92 

53.01 

59.68 

53.27 

80 

81 

61.13 

53.14 

60.90 

53.41 

60.67 

53.67 

60.43 

53.94 

81 

82 

61.89 

53.80 

61.65 

54.07 

61.41 

54.33 

61.18 

54.60 

82 

83 

62.64 

54.45 

62.40 

54,73 

62.16 

55.00 

61.92 

55.27 

83 

84 

63.40 

55.11 

63.15 

55.38 

62.91 

55.66 

62.67 

55.93 

84 

85 

64.15 

55.76 

63.91 

56.04 

63.66 

56.32 

63.41 

56.60 

85 

86 

64.90 

56.42 

64.66 

56.70 

64.41 

56.99 

64.16 

57.27 

86 

87 

65.66 

57.08 

65.41 

57.36 

65.16 

57.65 

64.91 

57.93 

87 

88 

66.41 

57.73 

66.16 

58.02 

65.91 

58.31 

65.65 

58.60 

88 

89 

67. 17 

53.39 

66.91 

58.68 

66.66 

58.97 

66.40 

59.26 

89 

90 

67.92 

59.05 

67.67 

59.34 

67.41 

59.64 

67.15 

59.93 

90 

91 

68.68 

59.70 

68.42 

60.00 

68.15 

60.30 

67.89 

60.60 

91 

92 

69.43 

60.36 

69.17 

60.66 

68.90 

60.96 

68.64 

61.26 

92 

93 

70. 19 

61.01 

69.92 

61.32 

69.65 

61.62 

69.38 

61.93 

93 

94 

70.94 

61.67 

70.67 

61.98 

70.40 

62.29 

70.13 

62.59 

94 

95 

71.70 

62.33 

71.43 

62.64 

71.15 

62.95 

70.88 

63.26 

95 

96 

72.45 

62.98 

72.18 

63.30 

71.90 

63.61 

71 .62 

63.92 

96 

97 

73.21 

63.64 

72.93 

63.96 

72.65 

64.27 

72.37 

64.59 

97 

98 

73.96 

64.29 

73.68 

64.62 

73.40 

64.94 

73.11 

65.26 

98 

99 

74.72 

64.95 

74.43 

65.28 

74.15 

65.60 

73.86 

65.92 

99 

100 

75.47 

65. 6L 

75.18 

65.93 

74.90 

66.26 

74.61 

66.59 

100 

® 

o 

3 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep. 

Lat. 

a> 

a 

a 

J3 

<n 

Q 

49  Deg. 

48f  Deg. 

48$  Deg. 

48£  Deg. 

.2 

P 

156 


TRAVERSE  TARLE. 


o 

42  Deg. 

42k  Deg. 

42£  Deg. 

42|  Deg. 

a 

35* 

p 

a 

o 

(S 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

1 

0.74 

0.67 

0.74 

0.67 

0.74 

0.68 

0.73 

0.68 

i 

2 

1.49 

1.34 

1.48 

1.34 

1.47 

1.35 

1.47 

1.36 

2 

3 

2.23 

2.01 

2.22 

2.02 

2.21 

2.03 

2.20 

2.04 

3 

4 

2.97 

2.68 

2.96 

2.69 

2.95 

2.70 

2.94 

2.72 

4 

5 

3.72 

3.35 

3.70 

3.36 

3.69 

3.38 

3.67 

3.39 

5 

6 

4.46 

4.01 

4.44 

4.03 

4.42 

4.05 

4.41 

4.07 

6 

7 

5.20 

4.68 

5.18 

4.71 

5.16 

4.73 

5.14 

4.75 

7 

8 

5.95 

5.35 

5.92 

5.38 

5.90 

5.40 

5.87 

5.43 

8 

9 

6.69 

6.02 

6.66 

6.05 

6.64 

6.08 

6.61 

6.11 

9 

10 

7.43 

6.69 

7.40 

6.72 

7.37 

6.76 

7.34 

6.79 

10 

11 

8.17 

7.36 

8.14 

7.40 

8.11 

7.43 

8.08 

7.47 

11 

12 

8.92 

8.03 

8.88 

8.07 

8.85 

8.11 

8.81 

8. 15 

12 

13 

9.66 

8.70 

9.62 

8.74 

9.58 

8.78 

9.55 

8.82 

13 

14 

10.40 

9.37 

10.36 

9.41 

10.32 

9.46 

10.28 

9.50 

14 

16 

11.15 

10.04 

11.10 

10.09 

11.06 

10.13 

11.01 

10.  IS 

15 

16 

11.89 

10.71 

11.84 

10.76 

11.80 

10.81 

11.75 

10.86 

16 

17 

12.63 

11.38 

12.58 

11.43 

12.53 

11.48 

12.48 

11.54 

17 

18 

13.38 

12.04 

13.32 

12. 10 

13.27 

12.16 

13.22 

12.22 

18 

19 

14.12 

12.71 

14.06 

12.77 

14.01 

12.84 

13.95 

12.90 

19 

20 

14.86 

13.38 

14.80 

13.45 

14.75 

13.51 

14.69 

13.58 

20 

21 

15.61 

14.05 

15.54 

14.12 

15.48 

14.19 

15.42 

14.25 

21 

22 

16.35 

14.72 

16.28 

14.79 

16.22 

14.86 

16.16 

14.93 

22 

23 

17.09 

15.39 

17.02 

15.46 

16.96 

15.54 

16.89 

15.61 

23 

24 

17.84 

16.06 

17.77 

16.14 

17.69 

16.21 

17.62 

16.29 

24 

25 

18.58 

16.73 

18.51 

16.81 

18.43 

16.89 

18.36 

16.97 

25 

26 

19.32 

17.40 

19.25 

17.48 

19.17 

17.57 

19.09 

17.65 

26 

27 

20.06 

18.07 

19.99 

18.15 

19.91 

18,24 

19.83 

18.33 

27 

2S 

20.81 

18.74 

20.73 

18.83 

20.64 

18.92 

20.56 

19.01 

28 

29 

21.55 

19.40 

21.47 

19.50 

21.38 

19.59 

21.30 

19.69 

29 

30 

22.29 

20.07 

22.21 

20.17 

22. 12 

20.27 

22.03 

20.36 

30 

31 

23.04 

20.74 

22.95 

20.84 

22.86 

20.94 

22.76 

21.04 

31 

32 

23.78 

21.41 

23.69 

21.52 

23.59 

21.62 

23.50 

21.72 

32 

33 

24.52 

22.08 

24.43 

22. 19 

24.33 

22.29 

24.23 

22.40 

33 

34 

25.27 

22.75 

25.17 

22.86 

25.07 

22.97 

24.97 

23.08 

34 

35 

26.01 

23.42 

25.91 

23.53 

25.80 

23.65 

25.70 

23.76 

35 

36 

26.75 

24.09 

26.65 

24.21 

26.54 

24.32 

26.44 

24.44 

36 

37 

27  50 

24.76 

27.39 

24. 8S 

27.28 

25.00 

27.17 

25.12 

37 

38 

28.24 

25.43 

28.13 

25.55 

28.02 

25.67 

27.90 

25.79 

38 

39 

28.98 

26.10 

28.87 

26.22 

28.75 

26.35 

28.64 

26.47 

39 

40 

29.73 

26.77 

29.61 

26.89 

29.49 

27.02 

29.37 

27.15 

40 

41 

30.47 

27.43 

30.35 

27.57 

30.23 

27.70 

30.11 

27.83 

41 

42 

31.21 

28.10 

31.09 

28.24 

30.97 

28.37 

30.84 

28.51 

42 

43 

31.96 

28.77 

31.83 

28.91 

31.70 

29.05 

31.58 

29.19 

43 

44 

32.70 

29.44 

32.57 

29.58 

32.44 

29.73 

132.31 

29.87 

44 

45 

33.44 

30.11 

33.31 

30.26 

33.18 

30.40 

j33.04 

30.55 

45 

46 

34.18 

30.78 

34.05 

30.93 

33.91 

31.08 

33.78 

31.22 

46 

47 

34.93 

31.45 

34.79 

31.60 

34.65 

31.75 

34.51 

31.90 

47 

48 

35.67 

32.12 

35.53 

32.27 

35.39 

32.43 

35.25 

32.58 

48 

49 

36.41 

32.79 

36.27 

32.95 

36.13 

33.10 

35,98 

33.26 

49 

50 

37.16 

33.46 

37.01 

33.62 

36.86 

33.78 

36.72 

33.94 

50 

3 

a 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

© 

o 

c 

1 Dista 

48  Deg. 

47 J Dog. 

47  i Deg. 

47* 

Deg. 

d 

U3 

Q 

TKAVEKBE  TABLE 


157 


g 

CO 

42  Deg. 

42*  Deg. 

42£  Deg. 

42|  Deg. 

o 

f 

3 

o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

O 

51 

37.90 

34.13 

37.75 

34.29 

37.60 

34.46 

37.45 

34.62 

"51 

52 

38.64 

34.79 

38.49 

34.96 

38.34 

35.13 

38.18 

35.30 

52 

53 

39.39 

35.46 

39.23 

35.64 

39.08 

35.81 

38.92 

35.98 

53 

54 

40.13 

36.13 

39.97 

36.31 

39.81 

36.48 

39.65 

36.66 

54 

55 

40.87 

36.80 

40.71 

36.98 

40.56 

37.16 

40.39 

37.33 

55 

56 

41.62 

37.47 

41.45 

37.65 

41.29 

37.83 

41.12 

38.01 

56 

57 

42.36 

38.14 

42.19 

38.32 

42.02 

38.51 

41.86 

38.69 

57 

58 

43.10 

33.81 

42.93 

39.00 

42.76 

39.18 

42.59 

39.37 

58 

59 

43.85 

39.48 

43.67 

39.67 

43.50 

39.86 

43.32 

40.05 

59 

60 

44.59 

40.15 

44.41 

40.34 

44.24 

40.54 

44.06 

40.73 

60 

61 

45.33 

40.82 

45.15 

41.01 

44.97 

41.21 

44.79 

41.41 

61 

62 

46.07 

41.49 

45.89 

41.69 

45.71 

41.89 

45,53 

42.09 

62 

63 

46.82 

42.16 

46.63 

42.36 

46.45 

42.56 

46.26 

42.76 

63 

64 

47.56 

42.82 

47.37 

43.03 

47.19 

43.24 

47.00 

43.44 

64 

65 

48.30 

43.49 

48.11 

43.70 

47.92 

43.91 

47.73 

44.12 

65 

66 

49.05 

44.16 

48.85 

44.38 

48.66 

44.59 

48.47 

44.80 

66 

67 

49.79 

44.83 

49.59 

45.05 

49.40 

45.26 

49.20 

45.48 

67 

68 

50.53 

45.50 

50.33 

45.72 

50.13 

45.94 

49.93 

46.16 

68 

69 

51.28 

46.17 

51.07 

46.39 

60.87 

46.62 

50.67 

46.84 

69 

70 

52.02 

46.84 

51.82 

47.07 

51.61 

47.29 

51.40 

47.52 

70 

71 

52.76 

47.51 

52.56 

47.74 

52.35 

47.97 

52.14 

48.19 

71 

72 

53.51 

48.18 

53.30 

48.41 

53.08 

48.64 

52.87 

48.87 

72 

73 

54.25 

48.85 

54.04 

49.08 

53.82 

49.32 

53.61 

49.55 

73 

74 

54.99 

49.52 

54.78 

49.76 

54.56 

49.99 

54.34 

50.23 

74 

75 

55.74 

50.18 

55.52 

50.43 

55.30 

50.67 

55.07 

50.91 

75 

76 

56.48 

50.85 

56.26 

51.10 

56.03 

51.34 

55.81 

51.59 

76 

77 

57.22 

51.62 

57.00 

51.77 

56.77 

52.02 

56.54 

52.27 

77 

78 

57.97 

52.  J 9 

57.74 

52.44 

57.51 

52.70 

57.28 

52.95 

78 

79 

58.71 

52.86 

58.48 

53.12 

58.24 

53.37 

58.01 

53.63 

79 

80 

59.45 

53.53 

59.22 

53.79 

58. 9S 

54.05 

58.75 

54.30 

80 

81 

60.19 

54.20 

59  96 

54.46 

59.72 

54.72 

59*48 

54.98 

81 

82 

60.94 

54.87 

60.  K) 

55.13 

60.46 

55.40 

60.21 

55.66 

82 

83 

61.68 

55.54 

61.44 

55.81 

61.19 

56.07 

60.95 

56.34 

83 

84 

62.42 

56.21 

62.18 

56.48 

61.93 

56.75 

61.68 

57.02 

84 

85 

63.17 

56.88 

62.92 

57. 15 

62.67 

57.43 

62.42 

57.70 

85 

86 

63.91 

57.55 

63.66 

57.82 

63.41 

58.10 

63.15 

58.38 

86 

87 

64.65 

58.21 

64.40 

58.50 

64.14 

58.78 

63.89 

59.06 

87 

88 

65.40 

58.88 

65.14 

59.17 

64.88 

59.45 

64.62 

59.73 

88 

89 

66.14 

59.55 

65.88 

59.84 

65.62 

60.13 

65.35 

60.41 

89 

90 

66.88 

60.22 

66.62 

60.51 

66.35 

60.80 

66.09 

61.09 

90 

91 

67.63 

60.89 

67.36 

61.19 

67.09 

61.48 

66.82 

61.77 

91 

92 

68.37 

61.56 

68.10 

61.86 

67.83 

62.15 

67.56 

62.45 

92 

93 

69.11 

62.23 

68.84 

62.53 

68.57 

62.83 

68.29 

63.13 

93 

94 

69.86 

62.90 

69.58 

63.20 

69.30 

63.51 

69.03 

63.81 

94 

95 

70.60 

63.57 

70.32 

63.87 

70.04 

64.18 

69.76 

64.49 

95 

96 

71.34 

64.24 

71.06 

64.55 

70.78 

64.86 

70.49 

65.16 

96 

97 

72.08 

64.91 

71.80 

65.22 

71.62 

65.53 

71.23 

65.84 

97 

98 

72.83 

65.57 

72.54 

65.89 

72.25 

66.21 

71.96 

66.52 

98 

99 

73.57 

66.24 

73.28 

66.66 

72.99 

66.88 

72.70 

67.20 

99 

100 

74.31 

66.91 

74.02 

67.24 

73.73 

67.56 

73.43 

67.88 

100 

<u 

o 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Distance.) 

.2 

Q 

48  Deg. 

47  J Deg. 

47$  Deg. 

47*  Deg. 

158 


TRAVERSE  TARLE. 


0 

5* 

1 
8 

43  Deg. 

43$  Deg. 

43$  Deg. 

43$  Deg. 

| Distance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

0.73 

1.46 

2.19 

2.93 

3.66 

4.39 

6.12 

5.85 

6.58 

7.31 

0.68 

1.36 

2.05 

2.73 

3.41 

4.09 

4.77 

5.46 

6.14 

6.82 

0.73 

1 *46 
2.19 
2.91 
3.64 
4.37 
6.10 
6.83 
6.56 
7.28 

0.69 

1.37 

2.06 

2.74 

3.43 

4.11 

4.80 

5.48 

6.17 

6.85 

0.73 

1.45 

2.18 

2.90 

3.63 

4.35 

5.08 

5.80 

6.53 

7.25 

0.69 

1.38 

2.07 

2.75 

3.44 

4.13 

4.82 

5.51 

6.20 

6.88 

0.72 

1.44 

2.17 

2.89 

3.61 

4.33 

5.06 

5.78 

6.50 

7.22 

0.69 
1.38 
2.07 
2.77 
3.46 
4.15 
4.84 
5.53 
6.22 
6 92 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

8.04 

7.50 

8.01 

7.54 

7.98 

7.57 

7.95 

7.61 

11 

12 

8.78 

8. 18 

8.74 

8.22 

8.70 

8.26 

8.67 

8.30 

12 

13 

9.51 

8.87 

9.47 

8.91 

9.43 

8.95 

9,39 

8.99 

13 

14 

10.24 

9.55 

10.20 

9.59 

10.16 

9.64 

10.11 

9.68 

14 

15 

10.97 

10.23 

10.93 

10,28 

10.88 

10.33 

10.84 

10.37 

15 

16 

11.70 

10.91 

11.65 

10.96 

11.61 

11.01 

11.56 

11.06 

16 

17 

12.43 

11.59 

12.38 

11.65 

12.33 

11.70 

12.28 

11.76 

17 

18 

13.16 

12.28 

13.11 

12.33 

L3.06 

12.39 

13.00 

12.45 

18 

19 

13.90 

12.96 

13.84 

13.02 

13.78 

13.08 

13.72 

13.14 

19 

20 

14.63 

13.64 

14.57 

13.70 

14.51 

J3.77 

14.45 

13.83 

20 

21 

15.36 

14.32 

15.30 

14.39 

15.23 

14.46 

15.17 

14.52 

21 

22 

16.09 

15.00 

16.02 

15.07 

15.96 

15.14 

15.89 

15.21 

22 

23 

16.82 

15.69 

16.75 

15.76 

16.68 

15.83 

16.61 

15.90 

23 

24 

17.55 

16.37 

17.48 

16.44 

17.41 

16.52 

17.34 

16.60 

24 

25 

IS. 28 

17.05 

18.21 

17.13 

18.13 

17.21 

18.06 

17.29 

25 

26 

19.02 

17.73 

18.94 

17.81 

18.86 

17.90 

18.78 

iv.  98 

26 

27 

19.75 

18.41 

19.67 

18  50 

19.59 

18.59 

19.50 

18.67 

27 

28 

20.48 

19.10 

20.39 

19.19 

20.31 

19.27 

20.23 

19.36 

28 

29 

21.21 

19.78 

21.12 

19.87 

21.04 

19.96 

20.95 

20.05 

29 

30 

21.94 

20.46 

21.85 

20.56 

21.76 

20.65 

21.67 

20.75 

30 

31 

22.67 

21.14 

22.58 

21.24 

22.49 

21.34 

22.39 

21.44 

31 

32 

23.40 

21.82 

23.31 

21.93 

23.21 

22.03 

23.12 

22.13 

32 

33 

24.13 

22.51 

24.04 

22.61 

23.94 

22.72 

23.84 

22.82 

33 

34 

24.87 

23.19 

24.76 

23.30 

24.66 

23.40 

24.56 

23.51 

34 

35 

25.60 

23.87 

25.49 

23.98 

25.39 

24.09 

25.28 

24.20 

35 

36 

26.33 

24.55 

26.22 

24.67 

26.11 

24.78 

26.01 

24.89 

36 

37 

27.06 

25.23 

26.95 

25.35 

26.84 

26.47 

26.73 

25.59 

37 

38 

27.79 

25.92 

27.68 

26.04 

27.56 

26.16 

27.45 

26.28 

38 

39 

28.52 

26.60 

28.41 

26.72 

28.29 

26.85 

28.17 

26.97 

39 

40 

29.25 

27.28 

29.13 

27.41 

29.01 

27.53 

28.89 

27.66 

40 

41 

29.99 

27.96 

29.86 

28.09 

29.74 

28.22 

29.62 

28.35 

41 

42 

30.72 

28.64 

30.59 

28.78 

30.47 

28.91 

30.34 

29.04 

42 

43 

31.45 

29.33 

31.32 

29.48 

31.19 

29.60 

31.06 

29.74 

43 

44 

32.18 

30.01 

32.05 

30.15 

31.92 

30.29 

31.78 

30.43 

44 

45 

32.91 

30.69 

32.78 

30.83 

32.64 

30.98 

32.51 

31.12 

45 

46 

33.64 

31.37 

33.51 

31.52 

33.37 

31.66 

33.23 

31.81 

46 

47 

34.37 

32.05 

34.23 

32.20 

34.09 

32.35 

33.95 

32.50 

47 

48 

35.10 

32.74 

34.96 

32.89 

34.82 

33.04 

34.67 

33.19 

48 

49 

35.84 

33.42 

35.69 

33.57 

35.54 

33.73 

35.40 

33.88 

49 

60 

36.57 

34.10 

36.42 

34.26 

36.27 

34.42 

36.12 

34.58 

50 

© 

o 

G 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

© 

o 

G 

cd 

.2 

Q 

47  Deg. 

46$  Deg. 

46$  Deg. 

46$  Dog. 

r) 

VI 

Q 

TRAVERSE  TABLE. 


159 


o 

w 

43  Deg. 

43}  Deg. 

43|  Deg. 

43}  Deg. 

O 

to 

3 

o 

<0 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 

S 

51 

37.30 

34.78 

37.15 

34.94 

36.99 

35.11 

36.84 

35.27 

51 

52 

38.03 

35.46 

37.88 

35.63 

37.72 

35.79 

37.56 

35.96 

52 

53 

38.76 

36.15 

38.60 

36.31 

38.44 

36.48 

38.29 

36.65 

53 

54 

39.49 

36.83 

39.33 

37.00 

39.17 

37.17 

39.01 

37.34 

64 

55 

40.22 

37.51 

40.06 

37.69 

39.90 

37.86 

39.73 

38.03 

55 

56 

40.96 

38.19 

40.79 

38.37 

40.62 

38.55 

40.45 

38.72 

56 

67 

41.69 

38.87 

41.52 

39.06 

41.35 

39.24 

41.17 

39.42 

67 

58 

42.42 

39.56 

42.25 

39.74 

42.07 

39.92 

41.90 

40.11 

68 

59 

43.15 

40.24 

42.97 

40.43 

42.80 

40.61 

42.62 

40.80 

69 

60 

43.88 

40.92 

43.70 

41.11 

43.52 

41.30 

43  34 

41.49 

60 

61 

44.61 

41.60 

44.43 

41.80 

44.25 

41.99 

44.06 

42.18 

61 

62 

45.34 

42.28 

45.16 

42.48 

44.97 

42.68 

44.79 

42.87 

62 

63 

46.08 

42.97 

45.89 

43.17 

45.70 

43.37 

45.51 

43.57 

63 

64 

46.81 

43.65 

46.62 

43.85 

46.42 

44.05 

46.23 

44.26 

64 

65 

47.54 

44.33 

47.34 

44.54 

47.15 

44.74 

46.95 

44.95 

65 

66 

48.27 

45.01 

48.07 

45.22 

47.87 

45.43 

47.68 

45.64 

66 

67 

49.00 

45.69 

48.80 

45.91 

48.60 

46.12 

48.40 

46.33 

67 

68 

49.73 

46.38 

49.53 

46.59 

49.33 

46.81 

49.12 

47.02 

6S 

69 

50.46 

47.06 

50.26 

47.28 

50.05 

47.50 

49.84 

47.71 

69 

70 

51.19 

47.74 

50.99 

47.96 

50.78 

48.18 

50.57 

48.41 

70 

71 

51.93 

48.42 

51.71 

4S.65 

51.50 

48.87 

51.29 

49.10 

71 

72 

52.66 

49.10 

52.44 

49.33 

52.23 

49.56 

52.01 

49.79 

72 

73 

53.39 

49.79 

53.17 

50.02 

52.95 

50.25 

52.73 

50.48 

73 

74 

54.12 

50.47 

53.90 

50.70 

53.68 

50.94 

53.45 

51.17 

74 

75 

54.85 

51.15 

54.63 

51.39 

54.40 

51.63 

54.18 

51.86 

75 

76 

55.58 

51.83 

55.36 

52.07 

55.13 

52.31 

54.90 

52.55 

76 

77 

56.31 

52.51 

56.08 

52.76 

55.85 

53.00 

55.62 

53.25 

77 

78 

57.05 

53.20 

56.81 

53.44 

56.58 

53.69 

56.34 

53.94 

78 

79 

57.78 

53.88 

57.54 

54.13 

57.30 

54.38 

57.07 

54.63 

79 

80 

58.51 

54.56 

58.27 

54.81 

58.03 

55.07 

57.79 

55.32 

80 

81 

59.24 

55.24 

59.00 

55.50 

58.76 

55.76 

58.51 

56.01 

81 

82 

59.97 

55.92 

59.73 

50.18 

59.48 

56.45 

59.23 

56.70 

82 

83 

60.70 

50.61 

00.45 

56.87 

60.21 

57.13 

59.96 

57.40 

83 

84 

61.43 

57.29 

61.18 

57.56 

60.93 

57.82 

60.68 

58.09 

84 

85 

62.17 

57.97 

61.91 

58.24 

61.66 

58.51 

61.40 

58,78 

85 

86 

62.90 

58.65 

62.64 

58.93 

62.38 

59.20 

62.12 

59.47 

86 

87 

63.63 

59.33 

63.37 

59.61 

63.11 

59.89 

62.85 

60.16 

87 

88 

64.36 

60.02 

64.10 

60.30 

63.83 

60.58 

63.57 

60.85 

88 

89 

65.09 

60.70 

64.82 

60.98 

64.56 

61.26 

64.29 

61.54 

89 

90 

65.82 

61.38 

65.55 

61.67 

65.28 

61.95 

65.01 

62.24 

90 

91 

66.55 

62.06 

66.28 

62.35 

66.01 

62.64 

65.74 

62.93 

91 

92 

67. 2S 

62.74 

67.01 

03.04 

66.73 

63.33 

66.46 

63.62 

92 

93 

68.02 

63.43 

67.74 

63.72 

67.46 

64.02 

67.18 

64.31 

93 

94 

68.75 

64.11 

68.47 

04.41 

68.19 

64.71 

67.90 

65.00 

94 

95 

69.48 

64.79 

09.20 

05.09 

68.91 

65.39 

1 68.62 

65.69 

95 

96 

70.21 

65.47 

69.92 

65.78 

69.04 

66.08 

69.35 

66.39 

96 

97 

70.94 

66.15 

70.65 

66.46 

70.36 

66.77 

70.07 

67.08 

97 

98 

71.67 

06.84 

71.37 

67.15 

71.09 

67.46 

70.79 

67.77 

98 

99 

72.40 

67.52 

72.11 

67.83 

71.81 

68.15 

71.51 

68.46 

99 

100 

73.14 

68.20 

72.84 

68.52 

72.54 

08.  S4 

72.24 

69.15 

100 

6 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

d 

o 

C 

ci 

w 

3 

47  Deg. 

46} 

Deg. 

^6-2  Deg. 

46}  Deg. 

ri 

cn 

Q 

30 


160 


TRAVERSE  TABLE, 


5 

co 

44  Dog. 

44$  Deg. 

44$  Deg. 

44}  Deg. 

45  Deg. 

O 

C7)’ 

P 

3 

O 

<0 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

O 

O 

1 

0.72 

0.69 

0.72 

0.70 

0.71 

0.70 

0.71 

0.71 

0.71 

0.71 

l 

2 

1.44 

1.39 

1.43 

1 .40 

1.43 

1.40 

1.42 

1 .41 

1 .41 

1.41 

2 

3 

2.16 

2.08 

2.15 

2.09 

2.14 

2.10 

2.13 

2.11 

2.12 

2.12 

3 

4 

2.88 

2.78 

2.87 

2.79 

2.85 

2.80 

2.84 

2.82 

2.83 

2.83 

4 

5 

3.60 

3.47 

3.58 

3.49 

3.57 

3.50 

3.55 

3.52 

3.54 

3.54 

5 

6 

4.32 

4.17 

4.30 

4.19 

4.28 

4.21 

4.26 

4.22 

4.24 

4.24 

6 

7 

5.04 

4.86 

5.01 

4.88 

4.99 

4.91 

4.97 

4.93 

4.95 

4.95 

7 

8 

5.75 

5.56 

5.73 

5.58 

5.71 

5.61 

5.68 

5.63 

5.66 

5.66 

8 

9 

6.47 

6.25 

6.45 

6.28 

6.42 

6.31 

6.39 

6.34 

6.36 

6.36 

9 

10 

7.19 

6.95 

7.16 

6.98 

7.13 

7.01 

7.10 

7.04 

7.07 

7.07 

10 

11 

7.91 

7.64 

7.88 

7.68 

7.85 

7.71 

7.81 

7.74 

7.78 

7.78 

11 

12 

8.63 

8.34 

8.60 

8.37 

8.56 

8.41 

8.52 

8.45 

8.49 

8.49 

12 

13 

9.35 

9.03 

9.31 

9.07 

9.27 

9.11 

9.23 

9.15 

9.19 

9.19 

13 

14 

10.07 

9.73 

10.03 

9.77 

9.99 

9.81 

9.94 

9.86 

9.90 

9.90 

14 

15 

10.79 

10.42 

10.74  10.47 

10.70 

10.51 

10.65 

10.56 

10.61 

10.61 

15 

1G 

11.51 

11.11 

11.46 

11.16 

11.41 

11.2! 

11.36 

11.26 

11.31 

11.31 

16 

17112.23 

11.81 

12.18 

11.86 

12.13 

11.92 

12.07 

11.97 

12.02 

12.02 

17 

18 

12.95 

12.50 

12.89 

12.56 

12.84 

12.62 

12.78 

12.67 

12.73 

12.73 

18 

19 

13.67 

13.20 

13.61 

13.26 

13.55 

13.32 

13.49 

13.38 

13.43 

13.43 

19 

20 

14.39 

13.89 

14.33 

13.96 

14.26 

14.02 

14.20 

14.08 

14.14 

14.14 

20 

21 

15.11 

14.59 

15.04 

14.65 

14.98 

14.72 

14.91 

14.78 

14.85 

14.85 

21 

22 

15.83 

15.28 

15.76 

15.35 

15.69 

15.42 

15.62 

15.49 

15.56 

15.56 

22 

23 

16.54 

15.98 

16.47 

16.05 

16.40 

16.12 

16.33 

16.19 

16.26 

16.26 

23 

24 

17.26 

16.67 

17.19 

16.75 

17.12 

16.82 

17.04 

16.90 

16.97 

16.97 

24 

25 

17.98 

17.37 

17.91 

17.44 

17.83 

17.52 

17.75 

17.60 

17.68 

17.68 

25 

26 

18.70 

18.06 

18.62 

18.14 

18.54 

18.22 

18.46 

18.30 

18.38 

18.38 

26 

27 

19.42 

18.76 

19.34 

18.84 

19.26 

18.92 

19.17 

19.01 

19.09 

19.09 

27 

28 

20.14 

19.45 

20.06 

19.54 

19.97 

19.63 

19.89 

19.71 

19.80 

19.80 

28 

29 

20.86 

20.15 

20.77 

20.24 

20.68 

20.33 

20.60 

20.42 

20.51 

20.51 

29 

30 

21.58 

20.84 

21.49 

20.93 

21.40 

21.03 

21.31 

21.12 

21.21 

21.21 

30 

31 

22.30 

21.53 

22.21 

21.63 

22.  fl 

21.73 

22.02 

21.82 

21.92 

21.92 

31 

32 

23.02 

22.23 

22.92 

22.33 

22.82 

22.43 

22.73 

22.53 

22.63 

22.63 

32 

33 

23.74 

22.92 

23.64 

23.03 

23.54 

23.13 

,23.44 

23.23 

23.33 

23.33 

33 

34 

24.46 

23.62 

24.35 

23.72 

24.25 

23.83 

24.15 

23.94 

24.04 

24.04 

34 

35 

25.18 

24.31 

25.07 

24.42 

24.96 

24.53 

124.86 

24.64 

24.75 

24.75 

35 

36 

25.90 

25.01 

25.79 

25.12 

25.68 

25.23 

125.57 

25.34 

25.46 

25.46 

36 

37 

26.62 

25.70 

26.50 

25.82 

26.39 

25.93 

26.28 

26.05 

26.16 

26.16 

37 

38 

27.33 

26.40 

27.22 

26.52 

27.10 

26.63 

26.99 

26.75 

26.87 

26.87 

38 

39 

28.05 

27.09 

27.94 

27.21 

27.82 

27.34 

27.70 

27.46 

27.58 

27.58 

39 

40 

28.77 

27.79 

28.65 

27.91 

28.53 

28.04 

28.41 

28.16 

28.28 

28.28 

40 

41 

29.49 

28.48 

29.37 

28.61 

29.24 

28.74 

29.12 

28.86 

28.99 

28.99 

41 

42 

30.21 

29.18 

30.08 

29.31 

29.96 

29.44 

29.83 

29.57 

£9.70 

29.70 

42 

43 

30.93 

29.87 

30.80 

30.00 

30.67 

30.14 

30.54 

30.27 

30.41 

30.41 

43 

44 

31.65 

30.56 

31.52 

30.70 

31.38 

30.84 

31.25 

30.98 

31.11 

31.11 

44 

45 

32.37 

31.26 

32.23 

31.40 

32.10 

31.54 

31.96 

31.68 

31.82 

31.82 

45 

40 

33.09 

31.95 

32.95 

32.10 

32.81 

32.24 

32.67 

32.38 

32.53 

32.53 

46 

47 

33.81 

32.  G5 

33.67 

32.80 

33.52 

32.94 

33.38 

33.09 

33.23 

33.23 

47 

48 

34.53 

33.34 

34.38 

33.49 

34.24 

33.64 

34.09 

33.79 

33.94 

33.94 

48 

49 

35.25 

34.04 

35.10 

34.19 

34.95 

34.34 

34.80 

34.50 

34.65 

34.65 

49 

60 

35.97 

34.73 

35.82 

34.89 

35.66 

35.05 

35.61 

35.20 

35.36 

35.36 

50 

0) 

o 

C 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

<x> 

o 

a 

rt 

r/j 

S 

46  Deg. 

45  } Deg. 

45$  Dog. 

45}  Deg. 

45  Deg. 

d 

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s 

TRAVERSE  TABLE 


101 


y 

t/i 

44  Deg. 

44$  Deg. 

44i  Deg. 

44f  Deg.  J 

45  Deg. 

O 

i/> 

p 

3 

o 

© 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

O 

© 

51 

36.69 

35.43 

36.53 

35.59 

36.38 

35.75 

36.22 

35.90; 

36.06 

36.06 

~5\ 

52 

37.41 

36.12 

37.25 

36.29 

37.09 

36.45 

36.93 

36.61 

36.77 

36.77 

52 

53 

38.12 

36.82 

37.96 

36.98 

37.80 

37.15 

37.64 

37.31 

37.48 

37.48 

53 

54 

38.84 

37.51 

38.68 

37.68 

38.52 

37.85 

38.35 

38.02 

38.18 

38.18 

54 

55 

39.56 

38.21 

39.40 

38. 3S 

39.23 

38.55 

39.06 

38.72! 

38.89 

38.89 

55 

56 

40.28 

38.90 

40.11 

39.08 

39.94 

39.25 

39.77 

39 - 42  j 

39.60 

39.60 

66 

57 

41.00 

39.60 

40.83 

39.77 

40.66 

39.95 

40.48 

40 . 13 

40.31 

40.31 

57 

58 

41.72 

40.29 

41.55 

40.47 

41.37 

40.65 

41.19 

40.83 

41.01 

41.01 

58 

69 

42.44 

40.98 

42.26 

41.17 

42.08 

41.35 

41.90 

41.54 

41.72 

41.72 

59 

60 

43.16 

41.68 

42.98 

41.87 

42.79 

42.05 

42.61 

42.24 

42.43 

42.43 

60 

61 

43.88 

42.37 

43.69 

42.57 

43.51 

42.76 

43.32 

42.94 

43.13 

43.13 

61 

62 

44.60 

43.07 

44.41 

43.26 

44.22 

43.46 

44.03 

43.65 

43.84 

43.84 

62 

63 

45.32 

43.76 

45. 13 

43.96 

44.93 

44.16 

44.74 

44.35 

44.55 

44.55 

63 

64 

46.04 

44.46 

45.84 

44.66 

45.65 

44.86 

45.45 

45.06 

45.25 

45.25 

64 

65 

46.76 

45.15 

46.56 

45.36 

46.36 

45.56 

46.16 

45.76 

45.96 

45.96 

65 

66 

47.48 

45.85 

47.28 

46.05 

47.07 

46.26 

46.87 

46.46 

46.67 

46.67 

66 

67 

48.20 

46.54 

47.99 

46.75 

47.79 

46.96 

47.58 

47.17 

47.38 

47.38 

67 

68 

48.92 

47.24 

48.71 

47.45 

48.50 

47.66 

48.29 

47.87 

48.08 

48.08 

68 

69 

49.63 

47.93 

49.42 

48.15 

49.21 

48.36 

49.00 

48.58 

48.79 

48.79 

69 

70 

50.35 

48.63 

50.14 

48.85 

49.93 

49.06 

49.71 

49.28 

49.50 

49.50 

70 

71 

51.07 

49.32 

50.86 

49.54 

50.64 

49.76 

50.42 

49.98 

50.20 

50.20 

71 

72 

51.79 

50.02 

51.57 

50.24 

51.35 

50.47 

151.13 

50.69 

50.91 

50.91 

72 

73 

52.51 

50.71 

52.29 

50.94 

'52.07 

51.17 

51.84 

51.39 

51.62 

51.62 

73 

74 

53.23 

51.40 

53.01 

51.64 

52.78 

51.87 

'52.55 

52.10 

52.33 

52.33 

74 

75 

53.95i52. 10 

53.72 

52.33 

53.49 

52.57 

53.26 

52.80 

53.03 

53.03 

75 

76 

54.67 

52.79 

54.44 

53.03 

54.21 

53.27 

53.97 

53.51 

53.74 

53.74 

76 

77 

55.39 

53.49 

55.16 

53.73 

54.92 

53.97 

54.68 

54.21 

54.45 

54.45 

77 

78 

56.11 

54. 18 

55.87 

54.43 

55.63 

54.67 

!55.39 

54.91 

55.15 

55.15 

78 

79 

56.83 

54.88 

56.59 

55.13 

56.35 

55.37 

'56.10 

55.62 

55.86 

55.86 

79 

80 

57.55 

55.57 

57.30 

55.82 

57.06 

56.07 

56.81 

56.32 

56.57  56.57 

80 

81 

58.27 

56.27 

58.02 

56.52 

57.77 

56.77 

57.52 

57.03 

57.28 

57.28 

81 

82 

58.99 

56.96 

58.74 

57.22 

58.49 

57.47 

59. 24157. 73 

57.98 

57.98 

82 

83 

59.71 

57.66 

59.45 

57.92 

59.20 

58.18 

!58. 95158.43 

58.69 

58.69 

83 

84 

60.42 

58.35 

60.17 

58.61 

59.91 

58.88 

I59.66j59.14 

59. 40159.40 

84 

85 

61.14 

59.05 

60.89 

59.31 

60.63 

59.58 

60. 37  59.84 

60. 10160. 10 

85 

86 

61.86 

59.74 

61.60 

60.01 

61.34 

60.28 

|61.08  60.55 

60.81  60.81 

86 

87 

62.58 

60.44 

62.32 

60.71 

62.05 

60.98 

161.79  61.25 

6l.52i61.52 

87 

88 

63.30 

61.13 

63.03 

61.41 

62.77 

61.68 

!62 . 50 

61.95 

62.23 

62.23 

88 

89 

64.02 

61.82 

63.75 

62.10 

63.48 

62.38 

163.21 

62.66 

62.93 

62.93 

89 

90 

64.74 

62.52 

64.47 

62.80 

64.19 

63.08 

63.92 

63.36 

63.64 

63.64 

90 

91 

65.46 

63.21 

65.18 

63.50 

64.91 

63.78 

64.63 

64.07 

64.35 

64.35 

91 

92 

66.18 

63.91 

65.90 

64.20 

65.62 

64.48 

65.34 

64.77 

65.05 

65.05 

92 

93 

66.90 

64.60 

66.62 

64.89 

66.33 

65.18 

66.05 

65.47 

65.76 

65.76 

93 

94 

67.62 

65.30 

67.33 

65.59 

67.05 

65.89 

66.76 

66. 18 

66.47 

66.47 

94 

95 

68.34 

65.99 

68.05 

66.29 

67.76 

66.59 

67.47 

66.88 

67.18 

67.18 

95 

96 

69.06 

66.69 

68.76 

66.99 

68.47 

67.29 

68.18 

67.59 

67.88 

67.88 

96 

97 

69.78 

67.38 

69.48 

67.69 

69.19 

67.99 

68.89 

68.29 

68.59 

68.59 

97 

98 

70.60 

68.08 

70.20 

68.38 

69.90 

68.69 

69.60 

68.99 

69.30 

69.30 

98 

99 

71.21 

68.77 

70.91 

69.08 

70.61 

69.39 

70.31 

69.70 

70.00 

70.00 

99 

100 

71.93 

69.47 

71.63 

69.78 

71.33 

70.09 

71.02 

70.40 

70.71 

70.71 

100 

© 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

O) 

o 

3 

OS 

09 

5 

46  Deg. 

45|  Deg. 

45  h Deg. 

45$  Deg. 

45  Deg. 

R) 

In 

Q 

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Gray’s  Lessons  and  Manual.  (In  one  volume)  ....  2.16 

For  Advanced  Students,  Teachers,  and  Practical  Botanists. 

Gray’s  Manual  of  Botany.  (Flora)  . . . . . .1.62 

Coulter’s  Botany  of  the  Rocky  Mountains  . . . .1.62 

A flora  adapted  to  the  mountain  section  of  the  United  States. 

Gray  and  Coulter’s  Text-Book  of  Western  Botany  . . . 2 16 

Being  Gray’s  Lessons  and  Coulter’s  Manual  bound  in  one  volume. 

Gray’s  Structural  Botany  ........  2.00 

Goodale’s  Physiological  Botany  ......  2.00 

Dana’s  Plants  and  their  Children  ......  .65 

Herrick’s  Chapters  on  Plant  Life  ......  .60 

Steele’s  Fourteen  Weeks  in  Botany  . . . . . .1.00 

Wood’s  How  to  Study  Plants 1.00 

Same  as  Steele’s  Fourteen  Weeks  in  Botany,  with  added  chapters  on 
Physiological  and  Systematic  Botany. 

Wood’s  Lessons  in  Botany.  (Revised)  .....  .90 

Wood's  New  American  Botanist  and  Florist.  (Revised)  . . 1.75 

Wood’s  Descriptive  Botany  . . . . . . . 1 25 

Being  the  flora  of  the  New  American  Botanist  and  Florist. 

Wood’s  Class  Book  of  Botany  . 2.50 

A standard  work  for  Advanced  Classes  and  for  the  Student’s  Library. 

Youmans’s  First  Book  in  Botany  ......  .64 

Youmans’s  Descriptive  Botany  . . . . . . .1.20 

Bentley’s  Physiological  Botany  . . . . . . .1.20 

A sequel  to  Youmans’s  Descriptive  Botany. 

Willis’s  Practical  Flora  . 1.50 

A valuable  supplementary  aid  to  any  text-book  in  the  study  of  Botany. 


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Text-Books  in  Geology 


DANA'S  GEOLOGICAL  STORY  BRIEFLY  TOLD 

By  James  D.  Dana.  Cloth,  i2mo,  302  pages  . . . $1.15 

A new  edition  of  this  popular  work  for  beginners  in  the  study  and 
for  the  general  reader.  The  book  has  been  entirely  rewritten,  and 
improved  by  the  addition  of  many  new  illustrations  and  interesting 
descriptions  of  the  latest  phases  and  discoveries  of  the  science.  In 
contents  and  dress  it  is  an  attractive  volume  either  for  the  reader  or 
student. 

DANA’S  REVISED  TEXT-BOOK  OF  GEOLOGY 

Edited  by  William  North  Rice,  Ph.D.,  LL.D.,  Professor 
of  Geology,  Wesleyan  University.  Cloth,  i2mo,  482  pages.  $1.40 
This  is  the  standard  text-book  for  high  school  and  elementary  college 
work.  The  book  has  been  thoroughly  revised,  enlarged,  and  improved, 
while  the  general  and  distinctive  features  of  the  former  work  have  been 
preserved.  As  now  published,  it  combines  the  results  of  the  life  experi- 
ence and  observation  of  its  distinguished  author  with  the  latest  discoveries 
and  researches  in  the  science. 

DANA’S  MANUAL  OF  GEOLOGY 

Cloth,  8vo,  1087  pages.  1575  illustrations  ....  $5.00 

This  great  geological  thesaurus  was  thoroughly  revised  and  entirely 
rewritten  under  the  direct  supervision  of  its  author,  just  before  his  death. 
It  is  recognized  as  a standard  authority  in  both  Europe  and  America, 
and  is  used  as  a manual  of  instruction  and  reference  in  all  higher 
institutions  of  learning. 

LE  CONTE’S  COMPEND  OF  GEOLOGY 

By  Joseph  Le  Conte,  LL.D.  Cloth,  i2mo,  399  pages  . $1.20 

Designed  for  high  schools,  academies,  and  all  secondary  schools. 
In  the  revised  edition  of  this  well-known  and  popular  text-book,  the 
general  plan  and  arrangement  remain  the  same,  but  such  modifications 
and  additions  have  been  made  as  were  necessary  to  bring  the  work  up  to 
the  present  condition  of  the  science. 


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Astronomy 


NEWCOMB'S  ELEMENTS  OF  ASTRONOMY 

Cloth,  12mo,  240  pages.  Illustrated  . . $1.00 

By  Simon  Newcomb,  Ph.D.,  LL.D. , Late  Professor  of  Mathe- 
matics and  Astronomy,  Johns  Hopkins  University,  and  formerly 
Senior  Professor  of  Mathematics,  United  States  Navy. 

This  volume  has  been  prepared  for  use  in  High  Schools  and 
College  Preparatory  Schools.  The  facts  and  laws  of  the  science  have 
been  condensed  within  small  compass,  and  the  subject  is  so  presented 
that  but  little  of  formal  mathematics  is  necessary  in  its  study.  A brief 
history  of  astronomy  is  included,  with  a General  Index  for  convenient 
reference,  and  numerous  illustrations,  figures,  and  diagrams. 

TODD’S  NEW  ASTRONOMY 

Cloth,  12mo,  480  pages.  Illustrated  . . . . . $1.30 

By  David  P.  Todd,  M.A.,  Ph.D.,  Professor  of  Astronomy  and 
Director  of  the  Observatory,  Amherst  College. 

The  noteworthy  feature  which  distinguishes  this  from  other  text- 
books on  astronomy  is  the  practical  way  in  which  the  subject  is  taught, 
largely  by  laboratory  experiments  and  observation  methods.  By  laying 
more  stress  on  the  physical  than  on  the  mathematical  facts  of  astronomy 
the  author  has  made  the  book  deeply  interesting.  The  marvelous  dis- 
coveries of  astronomy  in  recent  years  are  all  described,  while  the 
numerous  original  and  ingeniously  devised  illustrations  and  diagrams 
form  an  important  feature  of  the  book. 

BOWEN’S  ASTRONOMY  BY  OBSERVATION 

Boards,  Quarto,  94  pages $1.00 

By  Eliza  A.  Bowen. 

This  book,  unique  in  its  form  and  character,  and  original  in  its 
methods,  is  the  work  of  a practical  teacher,  and  its  reasoning  always 
appeals  to  observation,  study,  and  thought.  Careful  directions  are 
given  when,  how,  and  where  to  find  the  heavenly  bodies  and  the 
student  is  assisted  in  his  search  by  helpful  star-maps. 

STEELE’S  POPULAR  ASTRONOMY 

Cloth,  12mo,  349  pages.  Illustrated  . . . . $1.00 

Revised  and  Brought  Down  to  Date  by  Mabel  Loomis  Todd, 
author  of  “ Corona  and  Coronet,”  “ Total  Eclipses  of  the  Sun,”  etc. 
This  is  a revision  of  Steele’s  Descriptive  Astronomy,  and  while 
it  preserves  all  the  highly  desirable  features  of  the  original  work  it 
constitutes  substantially  a new  book.  The  revision  incorporates  all  the 
changes  and  additions  made  necessary  by  the  rapid  advance  of  practical 
and  physical  astronomy  in  the  last  fifteen  years,  and  contains  a large 
number  of  excellent  illustrations  and  diagrams,  together  with  several 
color  plates  and  a system  of  star-maps. 


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A New  Astronomy 

BY 

DAVID  P.  TODD,  M.A.,  Ph.D. 

Professor  of  Astronomy  and  Director  of  the  Observatory,  Amherst  College. 


Cloth,  i2mo,  480  pages.  Illustrated  - - Price,  $1.30 


This  book  is  designed  for  classes  pursuing  the  study  in 
High  Schools,  Academies,  and  Colleges.  The  author’s 
long  experience  as  a director  in  astronomical  observatories 
and  in  teaching  the  subject  has  given  him  unusual  qualifi- 
cations and  advantages  for  preparing  an  ideal  text-book. 

The  noteworthy  feature  which  distinguishes  this  from 
other  text-books  on  Astronomy  is  the  practical  way  in 
which  the  subjects  treated  are  enforced  by  laboratory 
experiments  and  methods.  In  this  the  author  follows  the 
principle  that  Astronomy  is  preeminently  a science  of 
observation  and  should  be  so  taught. 

By  placing  more  importance  on  the  physical  than  on 
the  mathematical  facts  of  Astronomy  the  author  has  made 
every  page  of  the  book  deeply  interesting  to  the  student 
and  the  general  reader.  The  treatment  of  the  planets  and 
other  heavenly  bodies  and  of  the  law  of  universal  gravita- 
tion is  unusually  full,  clear,  and  illuminative.  The  mar- 
velous discoveries  of  Astronomy  in  recent  years,  and  the 
latest  advances  in  methods  of  teaching  the  science,  are 
all  represented. 

The  illustrations  are  an  important  feature  of  the  book. 
Many  of  them  are  so  ingeniously  devised  that  they  explain 
at  a glance  what  pages  of  mere  description  could  not  make 
clear.  

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Scientific  Memoir  Series 

Edited  isy  JOSEPH  S.  AMES,  Ph.D. 
Johns  Hopkins  University 


The  Free  Expansion  of  Gases.  Memoirs  by  Gay-Lussac,  Joule, 

and  Joule  and  T homson.  Edited  by  Dr.  J.  S.  Amks  . . $0.75 

Prismatic  and  Diffraction  Spectra.  Memoirs  by  Joseph  von 

Fraunhofer.  Edited  by  Dr.  J.  S.  Ames  ....  .60 

Rontgen  Rays.  Memoirs  by  Rontgen,  Stokes,  and  J.  J.  Thomson. 

Edited  by  Dr.  George  F.  Barker 60 

The  Modern  Theory  of  Solution.  Memoirs  by  Pfeffer,Van’t  Hoff, 

Arrhenius,  and  Raoult.  Edited  by  Dr.  H.  C.  Jones  . . 1.00 

The  Laws  of  Gases.  Memoirs  by  Boyle  and  Amagat.  Edited  by 

Dr.  Carl  Barus 75 

The  Second  Law  of  Thermodynamics.  Memoirs  by  Carnot, 

Clausius,  and  Thomson.  Edited  by  Dr.  W.  F.  Magie  . .90 

The  Fundamental  Laws  of  Electrolytic  Conduction.  Memoirs  by 
Faraday,  ITittorf,  and  Kohlrausch.  Edited  by  Dr.  IL  M. 
Goodwin  ..........  .75 

The  Effects  of  a Magnetic  Field  on  Radiation.  Memoirs  by 

Faraday,  Kerr,  and  Zeeman.  Edited  by  Dr.  E.  P.  Lewis  . .75 

The  Laws  of  Gravitation.  Memoirs  by  Newton,  Bouguer,  and 

Cavendish.  Edited  by  Dr.  A.  S.  Mackenzie  . . 1 00 

The  Wave  Theory  of  Light.  Memoirs  by  Huygens,  Young,  and 

Fresnel.  Edited  by  Dr.  Henry  Crew  . . . .1.00 

The  Discovery  of  Induced  Electric  Currents.  Vol.  I.  Memoirs 

by  Joseph  Henry.  Edited  by  Dr.  J.  S.  Ames  ...  .75 

The  Discovery  of  Induced  Electric  Currents.  Vol.  II.  Memoirs 

by  Michael  Faraday.  Edited  by  Dr.  J.  S.  Ames  ...  .75 


Stereochemistry.  Memoirs  by  Pasteur,  Le  Bel,  and  Van’t  Hoff, 
together  with  selections  from  later  memoirs  by  Wislicenus 
and  others.  Edited  by  Dr.  G.  M.  Richardson  . 

The  Expansion  of  Gases.  Memoirs  by  Gay-Lussac  and  Regnault, 
Edited  by  Prof.  W.  W.  Randall  ..... 

Radiation  and  Absorption.  Memoirs  by  Prevost,  Balfour  Stewart, 
Kirchhoff,  and  Kirchhoff  and  Bunsen.  Edited  by  Dr. 
DeWitt  B.  Brace 


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